Octave 3.8, jcobi/3

Percentage Accurate: 94.8% → 99.3%

Time: 8.5s

Alternatives: 25

Speedup: 2.6×

Specification

\[\alpha > -1 \land \beta > -1\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot 1\\ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{t\_0}}{t\_0}}{t\_0 + 1} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 1.0))))
   (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) t_0) t_0) (+ t_0 1.0))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    t_0 = (alpha + beta) + (2.0d0 * 1.0d0)
    code = (((((alpha + beta) + (beta * alpha)) + 1.0d0) / t_0) / t_0) / (t_0 + 1.0d0)
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
}

Python

def code(alpha, beta):
	t_0 = (alpha + beta) + (2.0 * 1.0)
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0)

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(beta * alpha)) + 1.0) / t_0) / t_0) / Float64(t_0 + 1.0))
end

MATLAB

function tmp = code(alpha, beta)
	t_0 = (alpha + beta) + (2.0 * 1.0);
	tmp = (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]

Initial Program: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot 1\\ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{t\_0}}{t\_0}}{t\_0 + 1} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 1.0))))
   (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) t_0) t_0) (+ t_0 1.0))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    t_0 = (alpha + beta) + (2.0d0 * 1.0d0)
    code = (((((alpha + beta) + (beta * alpha)) + 1.0d0) / t_0) / t_0) / (t_0 + 1.0d0)
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
}

Python

def code(alpha, beta):
	t_0 = (alpha + beta) + (2.0 * 1.0)
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0)

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(beta * alpha)) + 1.0) / t_0) / t_0) / Float64(t_0 + 1.0))
end

MATLAB

function tmp = code(alpha, beta)
	t_0 = (alpha + beta) + (2.0 * 1.0);
	tmp = (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(1 + \left(\beta + \alpha\right)\right) + 2\\ t_1 := \beta \cdot \beta - 4\\ t_2 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 7200000:\\ \;\;\;\;\frac{\frac{\left(\frac{\beta}{t\_1} + \frac{\beta}{2 + \beta}\right) - \frac{2}{t\_1}}{2 + \beta}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{t\_2}, 1 - \frac{1}{\beta}\right)}{t\_2}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ 1.0 (+ beta alpha)) 2.0))
        (t_1 (- (* beta beta) 4.0))
        (t_2 (+ (+ alpha beta) 2.0)))
   (if (<= beta 7200000.0)
     (/
      (/ (- (+ (/ beta t_1) (/ beta (+ 2.0 beta))) (/ 2.0 t_1)) (+ 2.0 beta))
      t_0)
     (/ (/ (fma beta (/ alpha t_2) (- 1.0 (/ 1.0 beta))) t_2) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (1.0 + (beta + alpha)) + 2.0;
	double t_1 = (beta * beta) - 4.0;
	double t_2 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 7200000.0) {
		tmp = ((((beta / t_1) + (beta / (2.0 + beta))) - (2.0 / t_1)) / (2.0 + beta)) / t_0;
	} else {
		tmp = (fma(beta, (alpha / t_2), (1.0 - (1.0 / beta))) / t_2) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(1.0 + Float64(beta + alpha)) + 2.0)
	t_1 = Float64(Float64(beta * beta) - 4.0)
	t_2 = Float64(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (beta <= 7200000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(beta / t_1) + Float64(beta / Float64(2.0 + beta))) - Float64(2.0 / t_1)) / Float64(2.0 + beta)) / t_0);
	else
		tmp = Float64(Float64(fma(beta, Float64(alpha / t_2), Float64(1.0 - Float64(1.0 / beta))) / t_2) / t_0);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(1.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta * beta), $MachinePrecision] - 4.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 7200000.0], N[(N[(N[(N[(N[(beta / t$95$1), $MachinePrecision] + N[(beta / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(beta * N[(alpha / t$95$2), $MachinePrecision] + N[(1.0 - N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.2e6

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{1 + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1 + \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      6. lower-+.f64 99.9

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right)} + 2 \cdot 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\alpha + \beta\right)}\right) + 2 \cdot 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      9. lower-+.f64 99.9

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2 \cdot 1}} \]

      11. metadata-eval 99.9

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2}} \]

   4. Applied rewrites 99.9%

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\beta + \alpha\right)\right) + 2}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      3. div-add N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} + \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      4. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\alpha + \beta\right) + 2 \cdot 1}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{1}{\left(\beta + \alpha\right) + \color{blue}{2}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      11. flip-+ N/A

          \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\frac{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2 \cdot 2}{\left(\beta + \alpha\right) - 2}}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      12. associate-/r/ N/A

          \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2 \cdot 2} \cdot \left(\left(\beta + \alpha\right) - 2\right)} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2 \cdot 2}, \left(\beta + \alpha\right) - 2, \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   6. Applied rewrites 99.9%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{{\left(\alpha + \beta\right)}^{2} - 4}, \left(\alpha + \beta\right) - 2, \frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   7. Taylor expanded in alpha around 0

      \[\leadsto \frac{\color{blue}{\frac{\left(\frac{\beta}{2 + \beta} + \frac{\beta}{{\beta}^{2} - 4}\right) - 2 \cdot \frac{1}{{\beta}^{2} - 4}}{2 + \beta}}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(\frac{\beta}{2 + \beta} + \frac{\beta}{{\beta}^{2} - 4}\right) - 2 \cdot \frac{1}{{\beta}^{2} - 4}}{2 + \beta}}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\frac{\beta}{2 + \beta} + \frac{\beta}{{\beta}^{2} - 4}\right) - 2 \cdot \frac{1}{{\beta}^{2} - 4}}}{2 + \beta}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\frac{\beta}{{\beta}^{2} - 4} + \frac{\beta}{2 + \beta}\right)} - 2 \cdot \frac{1}{{\beta}^{2} - 4}}{2 + \beta}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\frac{\beta}{{\beta}^{2} - 4} + \frac{\beta}{2 + \beta}\right)} - 2 \cdot \frac{1}{{\beta}^{2} - 4}}{2 + \beta}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\color{blue}{\frac{\beta}{{\beta}^{2} - 4}} + \frac{\beta}{2 + \beta}\right) - 2 \cdot \frac{1}{{\beta}^{2} - 4}}{2 + \beta}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\frac{\left(\frac{\beta}{\color{blue}{{\beta}^{2} - 4}} + \frac{\beta}{2 + \beta}\right) - 2 \cdot \frac{1}{{\beta}^{2} - 4}}{2 + \beta}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      7. unpow2 N/A

         \[\leadsto \frac{\frac{\left(\frac{\beta}{\color{blue}{\beta \cdot \beta} - 4} + \frac{\beta}{2 + \beta}\right) - 2 \cdot \frac{1}{{\beta}^{2} - 4}}{2 + \beta}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\frac{\beta}{\color{blue}{\beta \cdot \beta} - 4} + \frac{\beta}{2 + \beta}\right) - 2 \cdot \frac{1}{{\beta}^{2} - 4}}{2 + \beta}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\frac{\beta}{\beta \cdot \beta - 4} + \color{blue}{\frac{\beta}{2 + \beta}}\right) - 2 \cdot \frac{1}{{\beta}^{2} - 4}}{2 + \beta}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{\frac{\left(\frac{\beta}{\beta \cdot \beta - 4} + \frac{\beta}{\color{blue}{2 + \beta}}\right) - 2 \cdot \frac{1}{{\beta}^{2} - 4}}{2 + \beta}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\frac{\left(\frac{\beta}{\beta \cdot \beta - 4} + \frac{\beta}{2 + \beta}\right) - \color{blue}{\frac{2 \cdot 1}{{\beta}^{2} - 4}}}{2 + \beta}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\frac{\left(\frac{\beta}{\beta \cdot \beta - 4} + \frac{\beta}{2 + \beta}\right) - \frac{\color{blue}{2}}{{\beta}^{2} - 4}}{2 + \beta}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\left(\frac{\beta}{\beta \cdot \beta - 4} + \frac{\beta}{2 + \beta}\right) - \color{blue}{\frac{2}{{\beta}^{2} - 4}}}{2 + \beta}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      14. lower--.f64 N/A

          \[\leadsto \frac{\frac{\left(\frac{\beta}{\beta \cdot \beta - 4} + \frac{\beta}{2 + \beta}\right) - \frac{2}{\color{blue}{{\beta}^{2} - 4}}}{2 + \beta}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      15. unpow2 N/A

          \[\leadsto \frac{\frac{\left(\frac{\beta}{\beta \cdot \beta - 4} + \frac{\beta}{2 + \beta}\right) - \frac{2}{\color{blue}{\beta \cdot \beta} - 4}}{2 + \beta}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\frac{\beta}{\beta \cdot \beta - 4} + \frac{\beta}{2 + \beta}\right) - \frac{2}{\color{blue}{\beta \cdot \beta} - 4}}{2 + \beta}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      17. lower-+.f64 67.3

          \[\leadsto \frac{\frac{\left(\frac{\beta}{\beta \cdot \beta - 4} + \frac{\beta}{2 + \beta}\right) - \frac{2}{\beta \cdot \beta - 4}}{\color{blue}{2 + \beta}}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   9. Applied rewrites 67.3%

      \[\leadsto \frac{\color{blue}{\frac{\left(\frac{\beta}{\beta \cdot \beta - 4} + \frac{\beta}{2 + \beta}\right) - \frac{2}{\beta \cdot \beta - 4}}{2 + \beta}}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   if 7.2e6 < beta

   1. Initial program 79.7%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{1 + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1 + \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      6. lower-+.f64 79.7

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right)} + 2 \cdot 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\alpha + \beta\right)}\right) + 2 \cdot 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      9. lower-+.f64 79.7

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2 \cdot 1}} \]

      11. metadata-eval 79.7

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2}} \]

   4. Applied rewrites 79.7%

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\beta + \alpha\right)\right) + 2}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      3. div-add N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} + \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      4. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\alpha + \beta\right) + 2 \cdot 1}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{1}{\left(\beta + \alpha\right) + \color{blue}{2}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      11. flip-+ N/A

          \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\frac{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2 \cdot 2}{\left(\beta + \alpha\right) - 2}}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      12. associate-/r/ N/A

          \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2 \cdot 2} \cdot \left(\left(\beta + \alpha\right) - 2\right)} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2 \cdot 2}, \left(\beta + \alpha\right) - 2, \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   6. Applied rewrites 79.7%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{{\left(\alpha + \beta\right)}^{2} - 4}, \left(\alpha + \beta\right) - 2, \frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right) + \frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{\alpha \cdot \beta + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      5. div-add N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\frac{\alpha \cdot \beta}{\left(\alpha + \beta\right) + 2} + \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2}\right)} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      6. associate-+l+ N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha \cdot \beta}{\left(\alpha + \beta\right) + 2} + \left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{\beta \cdot \alpha}}{\left(\alpha + \beta\right) + 2} + \left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      8. associate-/l* N/A

         \[\leadsto \frac{\frac{\color{blue}{\beta \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}} + \left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \color{blue}{\frac{\alpha}{\left(\alpha + \beta\right) + 2}}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \color{blue}{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   8. Applied rewrites 99.9%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + {\left(\left(\alpha + \beta\right) + 2\right)}^{-1}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   9. Taylor expanded in beta around inf

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \color{blue}{1 - \frac{1}{\beta}}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \color{blue}{1 - \frac{1}{\beta}}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

       2. lower-/.f64 99.9

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, 1 - \color{blue}{\frac{1}{\beta}}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   11. Applied rewrites 99.9%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \color{blue}{1 - \frac{1}{\beta}}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7200000:\\ \;\;\;\;\frac{\frac{\left(\frac{\beta}{\beta \cdot \beta - 4} + \frac{\beta}{2 + \beta}\right) - \frac{2}{\beta \cdot \beta - 4}}{2 + \beta}}{\left(1 + \left(\beta + \alpha\right)\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, 1 - \frac{1}{\beta}\right)}{\left(\alpha + \beta\right) + 2}}{\left(1 + \left(\beta + \alpha\right)\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \frac{\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{t\_0}, \frac{\alpha + \beta}{t\_0} + {t\_0}^{-1}\right)}{t\_0}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)))
   (/
    (/ (fma beta (/ alpha t_0) (+ (/ (+ alpha beta) t_0) (pow t_0 -1.0))) t_0)
    (+ (+ 1.0 (+ beta alpha)) 2.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	return (fma(beta, (alpha / t_0), (((alpha + beta) / t_0) + pow(t_0, -1.0))) / t_0) / ((1.0 + (beta + alpha)) + 2.0);
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	return Float64(Float64(fma(beta, Float64(alpha / t_0), Float64(Float64(Float64(alpha + beta) / t_0) + (t_0 ^ -1.0))) / t_0) / Float64(Float64(1.0 + Float64(beta + alpha)) + 2.0))
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, N[(N[(N[(beta * N[(alpha / t$95$0), $MachinePrecision] + N[(N[(N[(alpha + beta), $MachinePrecision] / t$95$0), $MachinePrecision] + N[Power[t$95$0, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(1.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 94.1%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{1 + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. lift-+.f64 N/A

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1 + \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   4. associate-+r+ N/A

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

   5. lower-+.f64 N/A

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

   6. lower-+.f64 94.1

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right)} + 2 \cdot 1} \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\alpha + \beta\right)}\right) + 2 \cdot 1} \]

   8. +-commutative N/A

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

   9. lower-+.f64 94.1

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2 \cdot 1}} \]

   11. metadata-eval 94.1

       \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2}} \]

4. Applied rewrites 94.1%

   \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\beta + \alpha\right)\right) + 2}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   3. div-add N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} + \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   4. +-commutative N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\alpha + \beta\right) + 2 \cdot 1}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{1}{\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   8. +-commutative N/A

      \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   9. lift-+.f64 N/A

      \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   10. metadata-eval N/A

       \[\leadsto \frac{\frac{\frac{1}{\left(\beta + \alpha\right) + \color{blue}{2}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   11. flip-+ N/A

       \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\frac{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2 \cdot 2}{\left(\beta + \alpha\right) - 2}}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   12. associate-/r/ N/A

       \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2 \cdot 2} \cdot \left(\left(\beta + \alpha\right) - 2\right)} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   13. lower-fma.f64 N/A

       \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2 \cdot 2}, \left(\beta + \alpha\right) - 2, \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

6. Applied rewrites 94.1%

   \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{{\left(\alpha + \beta\right)}^{2} - 4}, \left(\alpha + \beta\right) - 2, \frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
7. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right) + \frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   4. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{\frac{\color{blue}{\alpha \cdot \beta + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   5. div-add N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{\alpha \cdot \beta}{\left(\alpha + \beta\right) + 2} + \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2}\right)} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   6. associate-+l+ N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha \cdot \beta}{\left(\alpha + \beta\right) + 2} + \left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   7. *-commutative N/A

      \[\leadsto \frac{\frac{\frac{\color{blue}{\beta \cdot \alpha}}{\left(\alpha + \beta\right) + 2} + \left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   8. associate-/l* N/A

      \[\leadsto \frac{\frac{\color{blue}{\beta \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}} + \left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   9. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \color{blue}{\frac{\alpha}{\left(\alpha + \beta\right) + 2}}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   11. lower-+.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \color{blue}{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

8. Applied rewrites 99.9%

   \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + {\left(\left(\alpha + \beta\right) + 2\right)}^{-1}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

9. Final simplification 99.9%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + {\left(\left(\alpha + \beta\right) + 2\right)}^{-1}\right)}{\left(\alpha + \beta\right) + 2}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
10. Add Preprocessing

Alternative 3: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \frac{\mathsf{fma}\left(\frac{\alpha}{t\_0}, \frac{\beta}{t\_0}, \frac{\frac{\left(\alpha + \beta\right) - -1}{t\_0}}{t\_0}\right)}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))))
   (/
    (fma (/ alpha t_0) (/ beta t_0) (/ (/ (- (+ alpha beta) -1.0) t_0) t_0))
    (+ (+ 1.0 (+ beta alpha)) 2.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	return fma((alpha / t_0), (beta / t_0), ((((alpha + beta) - -1.0) / t_0) / t_0)) / ((1.0 + (beta + alpha)) + 2.0);
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	return Float64(fma(Float64(alpha / t_0), Float64(beta / t_0), Float64(Float64(Float64(Float64(alpha + beta) - -1.0) / t_0) / t_0)) / Float64(Float64(1.0 + Float64(beta + alpha)) + 2.0))
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(alpha / t$95$0), $MachinePrecision] * N[(beta / t$95$0), $MachinePrecision] + N[(N[(N[(N[(alpha + beta), $MachinePrecision] - -1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 94.1%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{1 + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. lift-+.f64 N/A

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1 + \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   4. associate-+r+ N/A

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

   5. lower-+.f64 N/A

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

   6. lower-+.f64 94.1

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right)} + 2 \cdot 1} \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\alpha + \beta\right)}\right) + 2 \cdot 1} \]

   8. +-commutative N/A

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

   9. lower-+.f64 94.1

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2 \cdot 1}} \]

   11. metadata-eval 94.1

       \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2}} \]

4. Applied rewrites 94.1%

   \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\beta + \alpha\right)\right) + 2}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   3. div-add N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} + \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   4. +-commutative N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\alpha + \beta\right) + 2 \cdot 1}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{1}{\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   8. +-commutative N/A

      \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   9. lift-+.f64 N/A

      \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   10. metadata-eval N/A

       \[\leadsto \frac{\frac{\frac{1}{\left(\beta + \alpha\right) + \color{blue}{2}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   11. flip-+ N/A

       \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\frac{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2 \cdot 2}{\left(\beta + \alpha\right) - 2}}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   12. associate-/r/ N/A

       \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2 \cdot 2} \cdot \left(\left(\beta + \alpha\right) - 2\right)} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   13. lower-fma.f64 N/A

       \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2 \cdot 2}, \left(\beta + \alpha\right) - 2, \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

6. Applied rewrites 94.1%

   \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{{\left(\alpha + \beta\right)}^{2} - 4}, \left(\alpha + \beta\right) - 2, \frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
7. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right) + \frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   4. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{\frac{\color{blue}{\alpha \cdot \beta + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   5. div-add N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{\alpha \cdot \beta}{\left(\alpha + \beta\right) + 2} + \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2}\right)} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   6. associate-+l+ N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha \cdot \beta}{\left(\alpha + \beta\right) + 2} + \left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   7. *-commutative N/A

      \[\leadsto \frac{\frac{\frac{\color{blue}{\beta \cdot \alpha}}{\left(\alpha + \beta\right) + 2} + \left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   8. associate-/l* N/A

      \[\leadsto \frac{\frac{\color{blue}{\beta \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}} + \left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   9. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \color{blue}{\frac{\alpha}{\left(\alpha + \beta\right) + 2}}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   11. lower-+.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \color{blue}{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

8. Applied rewrites 99.9%

   \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + {\left(\left(\alpha + \beta\right) + 2\right)}^{-1}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + {\left(\left(\alpha + \beta\right) + 2\right)}^{-1}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   2. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\beta \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} + \left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + {\left(\left(\alpha + \beta\right) + 2\right)}^{-1}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\beta \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} + \left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + {\left(\left(\alpha + \beta\right) + 2\right)}^{-1}\right)}{\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\frac{\beta \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} + \left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + {\left(\left(\alpha + \beta\right) + 2\right)}^{-1}\right)}{\left(\alpha + \beta\right) + \color{blue}{2}}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   5. div-add N/A

      \[\leadsto \frac{\color{blue}{\frac{\beta \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2} + \frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + {\left(\left(\alpha + \beta\right) + 2\right)}^{-1}}{\left(\alpha + \beta\right) + 2}}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   6. *-commutative N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \beta}}{\left(\alpha + \beta\right) + 2} + \frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + {\left(\left(\alpha + \beta\right) + 2\right)}^{-1}}{\left(\alpha + \beta\right) + 2}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   7. associate-/l* N/A

      \[\leadsto \frac{\color{blue}{\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2}} + \frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + {\left(\left(\alpha + \beta\right) + 2\right)}^{-1}}{\left(\alpha + \beta\right) + 2}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\beta}{\left(\alpha + \beta\right) + 2}, \frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + {\left(\left(\alpha + \beta\right) + 2\right)}^{-1}}{\left(\alpha + \beta\right) + 2}\right)}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

10. Applied rewrites 99.9%

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\alpha}{2 + \left(\alpha + \beta\right)}, \frac{\beta}{2 + \left(\alpha + \beta\right)}, \frac{\frac{\left(\alpha + \beta\right) - -1}{2 + \left(\alpha + \beta\right)}}{2 + \left(\alpha + \beta\right)}\right)}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
11. Add Preprocessing

Alternative 4: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(1 + \left(\beta + \alpha\right)\right) + 2\\ t_1 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+99}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{t\_1} - \frac{-1}{t\_1}}{t\_1}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{t\_1}, 1 - \frac{1}{\beta}\right)}{t\_1}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ 1.0 (+ beta alpha)) 2.0)) (t_1 (+ (+ alpha beta) 2.0)))
   (if (<= beta 2e+99)
     (/ (/ (- (/ (fma alpha beta (+ alpha beta)) t_1) (/ -1.0 t_1)) t_1) t_0)
     (/ (/ (fma beta (/ alpha t_1) (- 1.0 (/ 1.0 beta))) t_1) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (1.0 + (beta + alpha)) + 2.0;
	double t_1 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 2e+99) {
		tmp = (((fma(alpha, beta, (alpha + beta)) / t_1) - (-1.0 / t_1)) / t_1) / t_0;
	} else {
		tmp = (fma(beta, (alpha / t_1), (1.0 - (1.0 / beta))) / t_1) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(1.0 + Float64(beta + alpha)) + 2.0)
	t_1 = Float64(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (beta <= 2e+99)
		tmp = Float64(Float64(Float64(Float64(fma(alpha, beta, Float64(alpha + beta)) / t_1) - Float64(-1.0 / t_1)) / t_1) / t_0);
	else
		tmp = Float64(Float64(fma(beta, Float64(alpha / t_1), Float64(1.0 - Float64(1.0 / beta))) / t_1) / t_0);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(1.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 2e+99], N[(N[(N[(N[(N[(alpha * beta + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] - N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(beta * N[(alpha / t$95$1), $MachinePrecision] + N[(1.0 - N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.9999999999999999e99

   1. Initial program 99.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{1 + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1 + \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      6. lower-+.f64 99.9

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right)} + 2 \cdot 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\alpha + \beta\right)}\right) + 2 \cdot 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      9. lower-+.f64 99.9

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2 \cdot 1}} \]

      11. metadata-eval 99.9

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2}} \]

   4. Applied rewrites 99.9%

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\beta + \alpha\right)\right) + 2}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + \color{blue}{1 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) - \left(\mathsf{neg}\left(1\right)\right) \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) - \color{blue}{-1} \cdot 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) - \color{blue}{-1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      7. div-sub N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} - \frac{-1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} - \frac{-1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   6. Applied rewrites 99.9%

      \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2} - \frac{-1}{\left(\alpha + \beta\right) + 2}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   if 1.9999999999999999e99 < beta

   1. Initial program 75.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{1 + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1 + \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      6. lower-+.f64 75.0

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right)} + 2 \cdot 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\alpha + \beta\right)}\right) + 2 \cdot 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      9. lower-+.f64 75.0

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2 \cdot 1}} \]

      11. metadata-eval 75.0

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2}} \]

   4. Applied rewrites 75.0%

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\beta + \alpha\right)\right) + 2}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      3. div-add N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} + \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      4. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\alpha + \beta\right) + 2 \cdot 1}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{1}{\left(\beta + \alpha\right) + \color{blue}{2}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      11. flip-+ N/A

          \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\frac{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2 \cdot 2}{\left(\beta + \alpha\right) - 2}}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      12. associate-/r/ N/A

          \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2 \cdot 2} \cdot \left(\left(\beta + \alpha\right) - 2\right)} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2 \cdot 2}, \left(\beta + \alpha\right) - 2, \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   6. Applied rewrites 75.0%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{{\left(\alpha + \beta\right)}^{2} - 4}, \left(\alpha + \beta\right) - 2, \frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right) + \frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{\alpha \cdot \beta + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      5. div-add N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\frac{\alpha \cdot \beta}{\left(\alpha + \beta\right) + 2} + \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2}\right)} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      6. associate-+l+ N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha \cdot \beta}{\left(\alpha + \beta\right) + 2} + \left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{\beta \cdot \alpha}}{\left(\alpha + \beta\right) + 2} + \left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      8. associate-/l* N/A

         \[\leadsto \frac{\frac{\color{blue}{\beta \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}} + \left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \color{blue}{\frac{\alpha}{\left(\alpha + \beta\right) + 2}}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \color{blue}{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   8. Applied rewrites 99.9%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + {\left(\left(\alpha + \beta\right) + 2\right)}^{-1}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   9. Taylor expanded in beta around inf

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \color{blue}{1 - \frac{1}{\beta}}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \color{blue}{1 - \frac{1}{\beta}}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

       2. lower-/.f64 99.9

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, 1 - \color{blue}{\frac{1}{\beta}}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   11. Applied rewrites 99.9%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \color{blue}{1 - \frac{1}{\beta}}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+99}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2} - \frac{-1}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}{\left(1 + \left(\beta + \alpha\right)\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, 1 - \frac{1}{\beta}\right)}{\left(\alpha + \beta\right) + 2}}{\left(1 + \left(\beta + \alpha\right)\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+99}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) - -1}{t\_0}}{t\_0}}{\left(1 + \left(\beta + \alpha\right)\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\frac{\alpha - -1}{\beta} + \alpha\right) - -1\right) - \frac{\alpha + 2}{\beta} \cdot \left(\alpha - -1\right)}{t\_0}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)))
   (if (<= beta 2e+99)
     (/
      (/ (/ (- (+ (+ alpha beta) (* beta alpha)) -1.0) t_0) t_0)
      (+ (+ 1.0 (+ beta alpha)) 2.0))
     (/
      (/
       (-
        (- (+ (/ (- alpha -1.0) beta) alpha) -1.0)
        (* (/ (+ alpha 2.0) beta) (- alpha -1.0)))
       t_0)
      (+ 3.0 (+ alpha beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 2e+99) {
		tmp = (((((alpha + beta) + (beta * alpha)) - -1.0) / t_0) / t_0) / ((1.0 + (beta + alpha)) + 2.0);
	} else {
		tmp = ((((((alpha - -1.0) / beta) + alpha) - -1.0) - (((alpha + 2.0) / beta) * (alpha - -1.0))) / t_0) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (alpha + beta) + 2.0d0
    if (beta <= 2d+99) then
        tmp = (((((alpha + beta) + (beta * alpha)) - (-1.0d0)) / t_0) / t_0) / ((1.0d0 + (beta + alpha)) + 2.0d0)
    else
        tmp = ((((((alpha - (-1.0d0)) / beta) + alpha) - (-1.0d0)) - (((alpha + 2.0d0) / beta) * (alpha - (-1.0d0)))) / t_0) / (3.0d0 + (alpha + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 2e+99) {
		tmp = (((((alpha + beta) + (beta * alpha)) - -1.0) / t_0) / t_0) / ((1.0 + (beta + alpha)) + 2.0);
	} else {
		tmp = ((((((alpha - -1.0) / beta) + alpha) - -1.0) - (((alpha + 2.0) / beta) * (alpha - -1.0))) / t_0) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (alpha + beta) + 2.0
	tmp = 0
	if beta <= 2e+99:
		tmp = (((((alpha + beta) + (beta * alpha)) - -1.0) / t_0) / t_0) / ((1.0 + (beta + alpha)) + 2.0)
	else:
		tmp = ((((((alpha - -1.0) / beta) + alpha) - -1.0) - (((alpha + 2.0) / beta) * (alpha - -1.0))) / t_0) / (3.0 + (alpha + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (beta <= 2e+99)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(beta * alpha)) - -1.0) / t_0) / t_0) / Float64(Float64(1.0 + Float64(beta + alpha)) + 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(alpha - -1.0) / beta) + alpha) - -1.0) - Float64(Float64(Float64(alpha + 2.0) / beta) * Float64(alpha - -1.0))) / t_0) / Float64(3.0 + Float64(alpha + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (alpha + beta) + 2.0;
	tmp = 0.0;
	if (beta <= 2e+99)
		tmp = (((((alpha + beta) + (beta * alpha)) - -1.0) / t_0) / t_0) / ((1.0 + (beta + alpha)) + 2.0);
	else
		tmp = ((((((alpha - -1.0) / beta) + alpha) - -1.0) - (((alpha + 2.0) / beta) * (alpha - -1.0))) / t_0) / (3.0 + (alpha + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 2e+99], N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(1.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] + alpha), $MachinePrecision] - -1.0), $MachinePrecision] - N[(N[(N[(alpha + 2.0), $MachinePrecision] / beta), $MachinePrecision] * N[(alpha - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.9999999999999999e99

   1. Initial program 99.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{1 + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1 + \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      6. lower-+.f64 99.9

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right)} + 2 \cdot 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\alpha + \beta\right)}\right) + 2 \cdot 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      9. lower-+.f64 99.9

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2 \cdot 1}} \]

      11. metadata-eval 99.9

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2}} \]

   4. Applied rewrites 99.9%

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\beta + \alpha\right)\right) + 2}} \]

   if 1.9999999999999999e99 < beta

   1. Initial program 75.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \frac{\frac{\color{blue}{\left(1 + \left(\alpha + \left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right)\right)\right) - \frac{\left(1 + \alpha\right) \cdot \left(2 + \alpha\right)}{\beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(1 + \left(\alpha + \left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right)\right)\right) - \frac{\left(1 + \alpha\right) \cdot \left(2 + \alpha\right)}{\beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\left(\alpha + \left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right)\right) + 1\right)} - \frac{\left(1 + \alpha\right) \cdot \left(2 + \alpha\right)}{\beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\left(\alpha + \left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right)\right) + 1\right)} - \frac{\left(1 + \alpha\right) \cdot \left(2 + \alpha\right)}{\beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      4. +-commutative N/A

         \[\leadsto \frac{\frac{\left(\color{blue}{\left(\left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right) + \alpha\right)} + 1\right) - \frac{\left(1 + \alpha\right) \cdot \left(2 + \alpha\right)}{\beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\left(\color{blue}{\left(\left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right) + \alpha\right)} + 1\right) - \frac{\left(1 + \alpha\right) \cdot \left(2 + \alpha\right)}{\beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      6. div-add-rev N/A

         \[\leadsto \frac{\frac{\left(\left(\color{blue}{\frac{1 + \alpha}{\beta}} + \alpha\right) + 1\right) - \frac{\left(1 + \alpha\right) \cdot \left(2 + \alpha\right)}{\beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\color{blue}{\frac{1 + \alpha}{\beta}} + \alpha\right) + 1\right) - \frac{\left(1 + \alpha\right) \cdot \left(2 + \alpha\right)}{\beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\frac{\color{blue}{1 + \alpha}}{\beta} + \alpha\right) + 1\right) - \frac{\left(1 + \alpha\right) \cdot \left(2 + \alpha\right)}{\beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\frac{\left(\left(\frac{1 + \alpha}{\beta} + \alpha\right) + 1\right) - \color{blue}{\left(1 + \alpha\right) \cdot \frac{2 + \alpha}{\beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\left(\frac{1 + \alpha}{\beta} + \alpha\right) + 1\right) - \color{blue}{\left(1 + \alpha\right) \cdot \frac{2 + \alpha}{\beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{\frac{\left(\left(\frac{1 + \alpha}{\beta} + \alpha\right) + 1\right) - \color{blue}{\left(1 + \alpha\right)} \cdot \frac{2 + \alpha}{\beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\left(\left(\frac{1 + \alpha}{\beta} + \alpha\right) + 1\right) - \left(1 + \alpha\right) \cdot \color{blue}{\frac{2 + \alpha}{\beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      13. lower-+.f64 86.0

          \[\leadsto \frac{\frac{\left(\left(\frac{1 + \alpha}{\beta} + \alpha\right) + 1\right) - \left(1 + \alpha\right) \cdot \frac{\color{blue}{2 + \alpha}}{\beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Applied rewrites 86.0%

      \[\leadsto \frac{\frac{\color{blue}{\left(\left(\frac{1 + \alpha}{\beta} + \alpha\right) + 1\right) - \left(1 + \alpha\right) \cdot \frac{2 + \alpha}{\beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   7. Applied rewrites 86.0%

      \[\leadsto \color{blue}{\frac{\frac{\left(\left(\frac{\alpha - -1}{\beta} + \alpha\right) - -1\right) - \frac{\alpha + 2}{\beta} \cdot \left(\alpha - -1\right)}{\left(\alpha + \beta\right) + 2}}{3 + \left(\alpha + \beta\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+99}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) - -1}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}{\left(1 + \left(\beta + \alpha\right)\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\frac{\alpha - -1}{\beta} + \alpha\right) - -1\right) - \frac{\alpha + 2}{\beta} \cdot \left(\alpha - -1\right)}{\left(\alpha + \beta\right) + 2}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+99}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) - -1}{t\_0}}{t\_0}}{\left(1 + \left(\beta + \alpha\right)\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\frac{1 + \alpha}{\beta} + \alpha\right) - -1\right) - \left(1 + \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 5\right)}{\beta}}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)))
   (if (<= beta 2e+99)
     (/
      (/ (/ (- (+ (+ alpha beta) (* beta alpha)) -1.0) t_0) t_0)
      (+ (+ 1.0 (+ beta alpha)) 2.0))
     (/
      (/
       (-
        (- (+ (/ (+ 1.0 alpha) beta) alpha) -1.0)
        (* (+ 1.0 alpha) (/ (fma 2.0 alpha 5.0) beta)))
       beta)
      t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 2e+99) {
		tmp = (((((alpha + beta) + (beta * alpha)) - -1.0) / t_0) / t_0) / ((1.0 + (beta + alpha)) + 2.0);
	} else {
		tmp = ((((((1.0 + alpha) / beta) + alpha) - -1.0) - ((1.0 + alpha) * (fma(2.0, alpha, 5.0) / beta))) / beta) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (beta <= 2e+99)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(beta * alpha)) - -1.0) / t_0) / t_0) / Float64(Float64(1.0 + Float64(beta + alpha)) + 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.0 + alpha) / beta) + alpha) - -1.0) - Float64(Float64(1.0 + alpha) * Float64(fma(2.0, alpha, 5.0) / beta))) / beta) / t_0);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 2e+99], N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(1.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] + alpha), $MachinePrecision] - -1.0), $MachinePrecision] - N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(2.0 * alpha + 5.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.9999999999999999e99

   1. Initial program 99.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{1 + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1 + \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      6. lower-+.f64 99.9

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right)} + 2 \cdot 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\alpha + \beta\right)}\right) + 2 \cdot 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      9. lower-+.f64 99.9

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2 \cdot 1}} \]

      11. metadata-eval 99.9

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2}} \]

   4. Applied rewrites 99.9%

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\beta + \alpha\right)\right) + 2}} \]

   if 1.9999999999999999e99 < beta

   1. Initial program 75.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{1 + \beta}}{2 + \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. lower-+.f64 84.0

         \[\leadsto \frac{\frac{\frac{1 + \beta}{\color{blue}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Applied rewrites 84.0%

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{1 + \beta}{2 + \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1 + \beta}{2 + \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1 + \beta}{2 + \beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)}} \]

   7. Applied rewrites 84.0%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\beta - -1}{\beta + 2}}{3 + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2}} \]

   8. Taylor expanded in beta around inf

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(\alpha + \left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right)\right)\right) - \frac{\left(1 + \alpha\right) \cdot \left(5 + 2 \cdot \alpha\right)}{\beta}}{\beta}}}{\left(\alpha + \beta\right) + 2} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(\alpha + \left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right)\right)\right) - \frac{\left(1 + \alpha\right) \cdot \left(5 + 2 \cdot \alpha\right)}{\beta}}{\beta}}}{\left(\alpha + \beta\right) + 2} \]

   10. Applied rewrites 85.9%

       \[\leadsto \frac{\color{blue}{\frac{\left(\left(\frac{1 + \alpha}{\beta} + \alpha\right) + 1\right) - \left(1 + \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 5\right)}{\beta}}{\beta}}}{\left(\alpha + \beta\right) + 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+99}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) - -1}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}{\left(1 + \left(\beta + \alpha\right)\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\frac{1 + \alpha}{\beta} + \alpha\right) - -1\right) - \left(1 + \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 5\right)}{\beta}}{\beta}}{\left(\alpha + \beta\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+99}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \left(\beta - -2\right) \cdot \left(\beta - -3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\frac{1 + \alpha}{\beta} + \alpha\right) - -1\right) - \left(1 + \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 5\right)}{\beta}}{\beta}}{\left(\alpha + \beta\right) + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.2e+99)
   (/
    (- (fma beta alpha (+ beta alpha)) -1.0)
    (*
     (+ (+ beta alpha) 2.0)
     (fma
      (- (fma 2.0 beta alpha) -5.0)
      alpha
      (* (- beta -2.0) (- beta -3.0)))))
   (/
    (/
     (-
      (- (+ (/ (+ 1.0 alpha) beta) alpha) -1.0)
      (* (+ 1.0 alpha) (/ (fma 2.0 alpha 5.0) beta)))
     beta)
    (+ (+ alpha beta) 2.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.2e+99) {
		tmp = (fma(beta, alpha, (beta + alpha)) - -1.0) / (((beta + alpha) + 2.0) * fma((fma(2.0, beta, alpha) - -5.0), alpha, ((beta - -2.0) * (beta - -3.0))));
	} else {
		tmp = ((((((1.0 + alpha) / beta) + alpha) - -1.0) - ((1.0 + alpha) * (fma(2.0, alpha, 5.0) / beta))) / beta) / ((alpha + beta) + 2.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.2e+99)
		tmp = Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) - -1.0) / Float64(Float64(Float64(beta + alpha) + 2.0) * fma(Float64(fma(2.0, beta, alpha) - -5.0), alpha, Float64(Float64(beta - -2.0) * Float64(beta - -3.0)))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.0 + alpha) / beta) + alpha) - -1.0) - Float64(Float64(1.0 + alpha) * Float64(fma(2.0, alpha, 5.0) / beta))) / beta) / Float64(Float64(alpha + beta) + 2.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.2e+99], N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / N[(N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(N[(2.0 * beta + alpha), $MachinePrecision] - -5.0), $MachinePrecision] * alpha + N[(N[(beta - -2.0), $MachinePrecision] * N[(beta - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] + alpha), $MachinePrecision] - -1.0), $MachinePrecision] - N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(2.0 * alpha + 5.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.19999999999999978e99

   1. Initial program 99.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)\right)}} \]

   4. Applied rewrites 95.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)\right)}} \]

   5. Taylor expanded in alpha around 0

      \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \color{blue}{\left(\alpha \cdot \left(5 + \left(\alpha + 2 \cdot \beta\right)\right) + \left(2 + \beta\right) \cdot \left(3 + \beta\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\color{blue}{\left(5 + \left(\alpha + 2 \cdot \beta\right)\right) \cdot \alpha} + \left(2 + \beta\right) \cdot \left(3 + \beta\right)\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \color{blue}{\mathsf{fma}\left(5 + \left(\alpha + 2 \cdot \beta\right), \alpha, \left(2 + \beta\right) \cdot \left(3 + \beta\right)\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\alpha + 2 \cdot \beta\right) + 5}, \alpha, \left(2 + \beta\right) \cdot \left(3 + \beta\right)\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\left(\alpha + 2 \cdot \beta\right) + \color{blue}{5 \cdot 1}, \alpha, \left(2 + \beta\right) \cdot \left(3 + \beta\right)\right)} \]

      5. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\alpha + 2 \cdot \beta\right) - \left(\mathsf{neg}\left(5\right)\right) \cdot 1}, \alpha, \left(2 + \beta\right) \cdot \left(3 + \beta\right)\right)} \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\left(\alpha + 2 \cdot \beta\right) - \color{blue}{\left(\mathsf{neg}\left(5 \cdot 1\right)\right)}, \alpha, \left(2 + \beta\right) \cdot \left(3 + \beta\right)\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\left(\alpha + 2 \cdot \beta\right) - \left(\mathsf{neg}\left(\color{blue}{5}\right)\right), \alpha, \left(2 + \beta\right) \cdot \left(3 + \beta\right)\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\alpha + 2 \cdot \beta\right) - \left(\mathsf{neg}\left(5\right)\right)}, \alpha, \left(2 + \beta\right) \cdot \left(3 + \beta\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\color{blue}{\left(2 \cdot \beta + \alpha\right)} - \left(\mathsf{neg}\left(5\right)\right), \alpha, \left(2 + \beta\right) \cdot \left(3 + \beta\right)\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(2, \beta, \alpha\right)} - \left(\mathsf{neg}\left(5\right)\right), \alpha, \left(2 + \beta\right) \cdot \left(3 + \beta\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - \color{blue}{-5}, \alpha, \left(2 + \beta\right) \cdot \left(3 + \beta\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \color{blue}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}\right)} \]

      13. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \color{blue}{\left(\beta + 2\right)} \cdot \left(3 + \beta\right)\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \left(\beta + \color{blue}{2 \cdot 1}\right) \cdot \left(3 + \beta\right)\right)} \]

      15. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \color{blue}{\left(\beta - \left(\mathsf{neg}\left(2\right)\right) \cdot 1\right)} \cdot \left(3 + \beta\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \left(\beta - \color{blue}{-2} \cdot 1\right) \cdot \left(3 + \beta\right)\right)} \]

      17. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \left(\beta - \color{blue}{-2}\right) \cdot \left(3 + \beta\right)\right)} \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \color{blue}{\left(\beta - -2\right)} \cdot \left(3 + \beta\right)\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \left(\beta - -2\right) \cdot \color{blue}{\left(\beta + 3\right)}\right)} \]

      20. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \left(\beta - -2\right) \cdot \left(\beta + \color{blue}{3 \cdot 1}\right)\right)} \]

      21. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \left(\beta - -2\right) \cdot \color{blue}{\left(\beta - \left(\mathsf{neg}\left(3\right)\right) \cdot 1\right)}\right)} \]

      22. distribute-lft-neg-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \left(\beta - -2\right) \cdot \left(\beta - \color{blue}{\left(\mathsf{neg}\left(3 \cdot 1\right)\right)}\right)\right)} \]

      23. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \left(\beta - -2\right) \cdot \left(\beta - \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)} \]

      24. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \left(\beta - -2\right) \cdot \color{blue}{\left(\beta - \left(\mathsf{neg}\left(3\right)\right)\right)}\right)} \]

      25. metadata-eval 95.2

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \left(\beta - -2\right) \cdot \left(\beta - \color{blue}{-3}\right)\right)} \]

   7. Applied rewrites 95.2%

      \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \left(\beta - -2\right) \cdot \left(\beta - -3\right)\right)}} \]

   if 2.19999999999999978e99 < beta

   1. Initial program 75.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{1 + \beta}}{2 + \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. lower-+.f64 84.0

         \[\leadsto \frac{\frac{\frac{1 + \beta}{\color{blue}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Applied rewrites 84.0%

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{1 + \beta}{2 + \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1 + \beta}{2 + \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1 + \beta}{2 + \beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)}} \]

   7. Applied rewrites 84.0%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\beta - -1}{\beta + 2}}{3 + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2}} \]

   8. Taylor expanded in beta around inf

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(\alpha + \left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right)\right)\right) - \frac{\left(1 + \alpha\right) \cdot \left(5 + 2 \cdot \alpha\right)}{\beta}}{\beta}}}{\left(\alpha + \beta\right) + 2} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(\alpha + \left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right)\right)\right) - \frac{\left(1 + \alpha\right) \cdot \left(5 + 2 \cdot \alpha\right)}{\beta}}{\beta}}}{\left(\alpha + \beta\right) + 2} \]

   10. Applied rewrites 85.9%

       \[\leadsto \frac{\color{blue}{\frac{\left(\left(\frac{1 + \alpha}{\beta} + \alpha\right) + 1\right) - \left(1 + \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 5\right)}{\beta}}{\beta}}}{\left(\alpha + \beta\right) + 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+99}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \left(\beta - -2\right) \cdot \left(\beta - -3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\frac{1 + \alpha}{\beta} + \alpha\right) - -1\right) - \left(1 + \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 5\right)}{\beta}}{\beta}}{\left(\alpha + \beta\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ t_1 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 47000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{t\_1 \cdot \left(\left(3 + \left(\beta + \alpha\right)\right) \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{t\_0}, 1 - \frac{1}{\beta}\right)}{t\_0}}{\left(1 + \left(\beta + \alpha\right)\right) + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)) (t_1 (+ (+ beta alpha) 2.0)))
   (if (<= beta 47000000.0)
     (/
      (- (fma beta alpha (+ beta alpha)) -1.0)
      (* t_1 (* (+ 3.0 (+ beta alpha)) t_1)))
     (/
      (/ (fma beta (/ alpha t_0) (- 1.0 (/ 1.0 beta))) t_0)
      (+ (+ 1.0 (+ beta alpha)) 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double t_1 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 47000000.0) {
		tmp = (fma(beta, alpha, (beta + alpha)) - -1.0) / (t_1 * ((3.0 + (beta + alpha)) * t_1));
	} else {
		tmp = (fma(beta, (alpha / t_0), (1.0 - (1.0 / beta))) / t_0) / ((1.0 + (beta + alpha)) + 2.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	t_1 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 47000000.0)
		tmp = Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) - -1.0) / Float64(t_1 * Float64(Float64(3.0 + Float64(beta + alpha)) * t_1)));
	else
		tmp = Float64(Float64(fma(beta, Float64(alpha / t_0), Float64(1.0 - Float64(1.0 / beta))) / t_0) / Float64(Float64(1.0 + Float64(beta + alpha)) + 2.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 47000000.0], N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / N[(t$95$1 * N[(N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(beta * N[(alpha / t$95$0), $MachinePrecision] + N[(1.0 - N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(1.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.7e7

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)\right)}} \]

   4. Applied rewrites 95.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)\right)}} \]

   if 4.7e7 < beta

   1. Initial program 79.7%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{1 + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1 + \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      6. lower-+.f64 79.7

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right)} + 2 \cdot 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\alpha + \beta\right)}\right) + 2 \cdot 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      9. lower-+.f64 79.7

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2 \cdot 1}} \]

      11. metadata-eval 79.7

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2}} \]

   4. Applied rewrites 79.7%

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\beta + \alpha\right)\right) + 2}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      3. div-add N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} + \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      4. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\alpha + \beta\right) + 2 \cdot 1}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{1}{\left(\beta + \alpha\right) + \color{blue}{2}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      11. flip-+ N/A

          \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\frac{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2 \cdot 2}{\left(\beta + \alpha\right) - 2}}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      12. associate-/r/ N/A

          \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2 \cdot 2} \cdot \left(\left(\beta + \alpha\right) - 2\right)} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2 \cdot 2}, \left(\beta + \alpha\right) - 2, \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   6. Applied rewrites 79.7%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{{\left(\alpha + \beta\right)}^{2} - 4}, \left(\alpha + \beta\right) - 2, \frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right) + \frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{\alpha \cdot \beta + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      5. div-add N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\frac{\alpha \cdot \beta}{\left(\alpha + \beta\right) + 2} + \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2}\right)} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      6. associate-+l+ N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha \cdot \beta}{\left(\alpha + \beta\right) + 2} + \left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{\beta \cdot \alpha}}{\left(\alpha + \beta\right) + 2} + \left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      8. associate-/l* N/A

         \[\leadsto \frac{\frac{\color{blue}{\beta \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}} + \left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \color{blue}{\frac{\alpha}{\left(\alpha + \beta\right) + 2}}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \color{blue}{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   8. Applied rewrites 99.9%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + {\left(\left(\alpha + \beta\right) + 2\right)}^{-1}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   9. Taylor expanded in beta around inf

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \color{blue}{1 - \frac{1}{\beta}}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \color{blue}{1 - \frac{1}{\beta}}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

       2. lower-/.f64 99.9

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, 1 - \color{blue}{\frac{1}{\beta}}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   11. Applied rewrites 99.9%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \color{blue}{1 - \frac{1}{\beta}}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 47000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, 1 - \frac{1}{\beta}\right)}{\left(\alpha + \beta\right) + 2}}{\left(1 + \left(\beta + \alpha\right)\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+99}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \left(\beta - -2\right) \cdot \left(\beta - -3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\alpha + \beta\right) + 2}}{\left(1 + \left(\beta + \alpha\right)\right) + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.2e+99)
   (/
    (- (fma beta alpha (+ beta alpha)) -1.0)
    (*
     (+ (+ beta alpha) 2.0)
     (fma
      (- (fma 2.0 beta alpha) -5.0)
      alpha
      (* (- beta -2.0) (- beta -3.0)))))
   (/
    (/ (- alpha -1.0) (+ (+ alpha beta) 2.0))
    (+ (+ 1.0 (+ beta alpha)) 2.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.2e+99) {
		tmp = (fma(beta, alpha, (beta + alpha)) - -1.0) / (((beta + alpha) + 2.0) * fma((fma(2.0, beta, alpha) - -5.0), alpha, ((beta - -2.0) * (beta - -3.0))));
	} else {
		tmp = ((alpha - -1.0) / ((alpha + beta) + 2.0)) / ((1.0 + (beta + alpha)) + 2.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.2e+99)
		tmp = Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) - -1.0) / Float64(Float64(Float64(beta + alpha) + 2.0) * fma(Float64(fma(2.0, beta, alpha) - -5.0), alpha, Float64(Float64(beta - -2.0) * Float64(beta - -3.0)))));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / Float64(Float64(alpha + beta) + 2.0)) / Float64(Float64(1.0 + Float64(beta + alpha)) + 2.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.2e+99], N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / N[(N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(N[(2.0 * beta + alpha), $MachinePrecision] - -5.0), $MachinePrecision] * alpha + N[(N[(beta - -2.0), $MachinePrecision] * N[(beta - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.19999999999999978e99

   1. Initial program 99.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)\right)}} \]

   4. Applied rewrites 95.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)\right)}} \]

   5. Taylor expanded in alpha around 0

      \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \color{blue}{\left(\alpha \cdot \left(5 + \left(\alpha + 2 \cdot \beta\right)\right) + \left(2 + \beta\right) \cdot \left(3 + \beta\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\color{blue}{\left(5 + \left(\alpha + 2 \cdot \beta\right)\right) \cdot \alpha} + \left(2 + \beta\right) \cdot \left(3 + \beta\right)\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \color{blue}{\mathsf{fma}\left(5 + \left(\alpha + 2 \cdot \beta\right), \alpha, \left(2 + \beta\right) \cdot \left(3 + \beta\right)\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\alpha + 2 \cdot \beta\right) + 5}, \alpha, \left(2 + \beta\right) \cdot \left(3 + \beta\right)\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\left(\alpha + 2 \cdot \beta\right) + \color{blue}{5 \cdot 1}, \alpha, \left(2 + \beta\right) \cdot \left(3 + \beta\right)\right)} \]

      5. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\alpha + 2 \cdot \beta\right) - \left(\mathsf{neg}\left(5\right)\right) \cdot 1}, \alpha, \left(2 + \beta\right) \cdot \left(3 + \beta\right)\right)} \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\left(\alpha + 2 \cdot \beta\right) - \color{blue}{\left(\mathsf{neg}\left(5 \cdot 1\right)\right)}, \alpha, \left(2 + \beta\right) \cdot \left(3 + \beta\right)\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\left(\alpha + 2 \cdot \beta\right) - \left(\mathsf{neg}\left(\color{blue}{5}\right)\right), \alpha, \left(2 + \beta\right) \cdot \left(3 + \beta\right)\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\alpha + 2 \cdot \beta\right) - \left(\mathsf{neg}\left(5\right)\right)}, \alpha, \left(2 + \beta\right) \cdot \left(3 + \beta\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\color{blue}{\left(2 \cdot \beta + \alpha\right)} - \left(\mathsf{neg}\left(5\right)\right), \alpha, \left(2 + \beta\right) \cdot \left(3 + \beta\right)\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(2, \beta, \alpha\right)} - \left(\mathsf{neg}\left(5\right)\right), \alpha, \left(2 + \beta\right) \cdot \left(3 + \beta\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - \color{blue}{-5}, \alpha, \left(2 + \beta\right) \cdot \left(3 + \beta\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \color{blue}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}\right)} \]

      13. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \color{blue}{\left(\beta + 2\right)} \cdot \left(3 + \beta\right)\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \left(\beta + \color{blue}{2 \cdot 1}\right) \cdot \left(3 + \beta\right)\right)} \]

      15. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \color{blue}{\left(\beta - \left(\mathsf{neg}\left(2\right)\right) \cdot 1\right)} \cdot \left(3 + \beta\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \left(\beta - \color{blue}{-2} \cdot 1\right) \cdot \left(3 + \beta\right)\right)} \]

      17. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \left(\beta - \color{blue}{-2}\right) \cdot \left(3 + \beta\right)\right)} \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \color{blue}{\left(\beta - -2\right)} \cdot \left(3 + \beta\right)\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \left(\beta - -2\right) \cdot \color{blue}{\left(\beta + 3\right)}\right)} \]

      20. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \left(\beta - -2\right) \cdot \left(\beta + \color{blue}{3 \cdot 1}\right)\right)} \]

      21. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \left(\beta - -2\right) \cdot \color{blue}{\left(\beta - \left(\mathsf{neg}\left(3\right)\right) \cdot 1\right)}\right)} \]

      22. distribute-lft-neg-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \left(\beta - -2\right) \cdot \left(\beta - \color{blue}{\left(\mathsf{neg}\left(3 \cdot 1\right)\right)}\right)\right)} \]

      23. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \left(\beta - -2\right) \cdot \left(\beta - \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)} \]

      24. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \left(\beta - -2\right) \cdot \color{blue}{\left(\beta - \left(\mathsf{neg}\left(3\right)\right)\right)}\right)} \]

      25. metadata-eval 95.2

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \left(\beta - -2\right) \cdot \left(\beta - \color{blue}{-3}\right)\right)} \]

   7. Applied rewrites 95.2%

      \[\leadsto \frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \left(\beta - -2\right) \cdot \left(\beta - -3\right)\right)}} \]

   if 2.19999999999999978e99 < beta

   1. Initial program 75.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{1 + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1 + \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      6. lower-+.f64 75.0

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right)} + 2 \cdot 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\alpha + \beta\right)}\right) + 2 \cdot 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      9. lower-+.f64 75.0

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2 \cdot 1}} \]

      11. metadata-eval 75.0

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2}} \]

   4. Applied rewrites 75.0%

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\beta + \alpha\right)\right) + 2}} \]

   5. Taylor expanded in beta around -inf

      \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \left(-1 \cdot \alpha - 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{neg}\left(\left(-1 \cdot \alpha - 1\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{-\left(-1 \cdot \alpha - 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\frac{-\color{blue}{\left(-1 \cdot \alpha - 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\frac{-\left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} - 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      5. lower-neg.f64 85.7

         \[\leadsto \frac{\frac{-\left(\color{blue}{\left(-\alpha\right)} - 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   7. Applied rewrites 85.7%

      \[\leadsto \frac{\frac{\color{blue}{-\left(\left(-\alpha\right) - 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+99}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \beta, \alpha\right) - -5, \alpha, \left(\beta - -2\right) \cdot \left(\beta - -3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\alpha + \beta\right) + 2}}{\left(1 + \left(\beta + \alpha\right)\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+99}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{t\_0 \cdot \left(\left(3 + \left(\beta + \alpha\right)\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\alpha + \beta\right) + 2}}{\left(1 + \left(\beta + \alpha\right)\right) + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 2.2e+99)
     (/
      (- (fma beta alpha (+ beta alpha)) -1.0)
      (* t_0 (* (+ 3.0 (+ beta alpha)) t_0)))
     (/
      (/ (- alpha -1.0) (+ (+ alpha beta) 2.0))
      (+ (+ 1.0 (+ beta alpha)) 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 2.2e+99) {
		tmp = (fma(beta, alpha, (beta + alpha)) - -1.0) / (t_0 * ((3.0 + (beta + alpha)) * t_0));
	} else {
		tmp = ((alpha - -1.0) / ((alpha + beta) + 2.0)) / ((1.0 + (beta + alpha)) + 2.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 2.2e+99)
		tmp = Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) - -1.0) / Float64(t_0 * Float64(Float64(3.0 + Float64(beta + alpha)) * t_0)));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / Float64(Float64(alpha + beta) + 2.0)) / Float64(Float64(1.0 + Float64(beta + alpha)) + 2.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 2.2e+99], N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / N[(t$95$0 * N[(N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.19999999999999978e99

   1. Initial program 99.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)\right)}} \]

   4. Applied rewrites 95.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)\right)}} \]

   if 2.19999999999999978e99 < beta

   1. Initial program 75.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{1 + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1 + \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      6. lower-+.f64 75.0

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right)} + 2 \cdot 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\alpha + \beta\right)}\right) + 2 \cdot 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      9. lower-+.f64 75.0

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2 \cdot 1}} \]

      11. metadata-eval 75.0

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2}} \]

   4. Applied rewrites 75.0%

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\beta + \alpha\right)\right) + 2}} \]

   5. Taylor expanded in beta around -inf

      \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \left(-1 \cdot \alpha - 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{neg}\left(\left(-1 \cdot \alpha - 1\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{-\left(-1 \cdot \alpha - 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\frac{-\color{blue}{\left(-1 \cdot \alpha - 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\frac{-\left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} - 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      5. lower-neg.f64 85.7

         \[\leadsto \frac{\frac{-\left(\color{blue}{\left(-\alpha\right)} - 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   7. Applied rewrites 85.7%

      \[\leadsto \frac{\frac{\color{blue}{-\left(\left(-\alpha\right) - 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+99}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\alpha + \beta\right) + 2}}{\left(1 + \left(\beta + \alpha\right)\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(1 + \left(\beta + \alpha\right)\right) + 2\\ \mathbf{if}\;\beta \leq 82:\\ \;\;\;\;\frac{\frac{\frac{1 + \alpha}{2 + \alpha}}{2 + \alpha}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\alpha + \beta\right) + 2}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ 1.0 (+ beta alpha)) 2.0)))
   (if (<= beta 82.0)
     (/ (/ (/ (+ 1.0 alpha) (+ 2.0 alpha)) (+ 2.0 alpha)) t_0)
     (/ (/ (- alpha -1.0) (+ (+ alpha beta) 2.0)) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (1.0 + (beta + alpha)) + 2.0;
	double tmp;
	if (beta <= 82.0) {
		tmp = (((1.0 + alpha) / (2.0 + alpha)) / (2.0 + alpha)) / t_0;
	} else {
		tmp = ((alpha - -1.0) / ((alpha + beta) + 2.0)) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + (beta + alpha)) + 2.0d0
    if (beta <= 82.0d0) then
        tmp = (((1.0d0 + alpha) / (2.0d0 + alpha)) / (2.0d0 + alpha)) / t_0
    else
        tmp = ((alpha - (-1.0d0)) / ((alpha + beta) + 2.0d0)) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (1.0 + (beta + alpha)) + 2.0;
	double tmp;
	if (beta <= 82.0) {
		tmp = (((1.0 + alpha) / (2.0 + alpha)) / (2.0 + alpha)) / t_0;
	} else {
		tmp = ((alpha - -1.0) / ((alpha + beta) + 2.0)) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (1.0 + (beta + alpha)) + 2.0
	tmp = 0
	if beta <= 82.0:
		tmp = (((1.0 + alpha) / (2.0 + alpha)) / (2.0 + alpha)) / t_0
	else:
		tmp = ((alpha - -1.0) / ((alpha + beta) + 2.0)) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(1.0 + Float64(beta + alpha)) + 2.0)
	tmp = 0.0
	if (beta <= 82.0)
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + alpha)) / Float64(2.0 + alpha)) / t_0);
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / Float64(Float64(alpha + beta) + 2.0)) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (1.0 + (beta + alpha)) + 2.0;
	tmp = 0.0;
	if (beta <= 82.0)
		tmp = (((1.0 + alpha) / (2.0 + alpha)) / (2.0 + alpha)) / t_0;
	else
		tmp = ((alpha - -1.0) / ((alpha + beta) + 2.0)) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(1.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 82.0], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 82

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{1 + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1 + \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      6. lower-+.f64 99.9

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right)} + 2 \cdot 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\alpha + \beta\right)}\right) + 2 \cdot 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      9. lower-+.f64 99.9

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2 \cdot 1}} \]

      11. metadata-eval 99.9

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2}} \]

   4. Applied rewrites 99.9%

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\beta + \alpha\right)\right) + 2}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      3. div-add N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} + \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      4. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\alpha + \beta\right) + 2 \cdot 1}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{1}{\left(\beta + \alpha\right) + \color{blue}{2}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      11. flip-+ N/A

          \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\frac{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2 \cdot 2}{\left(\beta + \alpha\right) - 2}}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      12. associate-/r/ N/A

          \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2 \cdot 2} \cdot \left(\left(\beta + \alpha\right) - 2\right)} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2 \cdot 2}, \left(\beta + \alpha\right) - 2, \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   6. Applied rewrites 99.9%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{{\left(\alpha + \beta\right)}^{2} - 4}, \left(\alpha + \beta\right) - 2, \frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right) + \frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{\alpha \cdot \beta + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      5. div-add N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\frac{\alpha \cdot \beta}{\left(\alpha + \beta\right) + 2} + \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2}\right)} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      6. associate-+l+ N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha \cdot \beta}{\left(\alpha + \beta\right) + 2} + \left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{\beta \cdot \alpha}}{\left(\alpha + \beta\right) + 2} + \left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      8. associate-/l* N/A

         \[\leadsto \frac{\frac{\color{blue}{\beta \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}} + \left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \color{blue}{\frac{\alpha}{\left(\alpha + \beta\right) + 2}}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \color{blue}{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   8. Applied rewrites 99.9%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + {\left(\left(\alpha + \beta\right) + 2\right)}^{-1}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   9. Taylor expanded in beta around 0

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2 + \alpha} + \frac{\alpha}{2 + \alpha}}{2 + \alpha}}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2 + \alpha} + \frac{\alpha}{2 + \alpha}}{2 + \alpha}}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

       2. div-add-rev N/A

          \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \alpha}{2 + \alpha}}}{2 + \alpha}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

       3. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \alpha}{2 + \alpha}}}{2 + \alpha}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

       4. lower-+.f64 N/A

          \[\leadsto \frac{\frac{\frac{\color{blue}{1 + \alpha}}{2 + \alpha}}{2 + \alpha}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

       5. lower-+.f64 N/A

          \[\leadsto \frac{\frac{\frac{1 + \alpha}{\color{blue}{2 + \alpha}}}{2 + \alpha}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

       6. lower-+.f64 99.3

          \[\leadsto \frac{\frac{\frac{1 + \alpha}{2 + \alpha}}{\color{blue}{2 + \alpha}}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   11. Applied rewrites 99.3%

       \[\leadsto \frac{\color{blue}{\frac{\frac{1 + \alpha}{2 + \alpha}}{2 + \alpha}}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   if 82 < beta

   1. Initial program 80.2%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{1 + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1 + \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      6. lower-+.f64 80.2

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right)} + 2 \cdot 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\alpha + \beta\right)}\right) + 2 \cdot 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      9. lower-+.f64 80.2

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2 \cdot 1}} \]

      11. metadata-eval 80.2

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2}} \]

   4. Applied rewrites 80.2%

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\beta + \alpha\right)\right) + 2}} \]

   5. Taylor expanded in beta around -inf

      \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \left(-1 \cdot \alpha - 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{neg}\left(\left(-1 \cdot \alpha - 1\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{-\left(-1 \cdot \alpha - 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\frac{-\color{blue}{\left(-1 \cdot \alpha - 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\frac{-\left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} - 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      5. lower-neg.f64 82.0

         \[\leadsto \frac{\frac{-\left(\color{blue}{\left(-\alpha\right)} - 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   7. Applied rewrites 82.0%

      \[\leadsto \frac{\frac{\color{blue}{-\left(\left(-\alpha\right) - 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 82:\\ \;\;\;\;\frac{\frac{\frac{1 + \alpha}{2 + \alpha}}{2 + \alpha}}{\left(1 + \left(\beta + \alpha\right)\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\alpha + \beta\right) + 2}}{\left(1 + \left(\beta + \alpha\right)\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 98.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 9.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{\frac{1 + \beta}{2 + \beta}}{3 + \beta}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{t\_0}}{\left(1 + \left(\beta + \alpha\right)\right) + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)))
   (if (<= beta 9.2e+14)
     (/ (/ (/ (+ 1.0 beta) (+ 2.0 beta)) (+ 3.0 beta)) t_0)
     (/ (/ (- alpha -1.0) t_0) (+ (+ 1.0 (+ beta alpha)) 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 9.2e+14) {
		tmp = (((1.0 + beta) / (2.0 + beta)) / (3.0 + beta)) / t_0;
	} else {
		tmp = ((alpha - -1.0) / t_0) / ((1.0 + (beta + alpha)) + 2.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (alpha + beta) + 2.0d0
    if (beta <= 9.2d+14) then
        tmp = (((1.0d0 + beta) / (2.0d0 + beta)) / (3.0d0 + beta)) / t_0
    else
        tmp = ((alpha - (-1.0d0)) / t_0) / ((1.0d0 + (beta + alpha)) + 2.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 9.2e+14) {
		tmp = (((1.0 + beta) / (2.0 + beta)) / (3.0 + beta)) / t_0;
	} else {
		tmp = ((alpha - -1.0) / t_0) / ((1.0 + (beta + alpha)) + 2.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (alpha + beta) + 2.0
	tmp = 0
	if beta <= 9.2e+14:
		tmp = (((1.0 + beta) / (2.0 + beta)) / (3.0 + beta)) / t_0
	else:
		tmp = ((alpha - -1.0) / t_0) / ((1.0 + (beta + alpha)) + 2.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (beta <= 9.2e+14)
		tmp = Float64(Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(3.0 + beta)) / t_0);
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / t_0) / Float64(Float64(1.0 + Float64(beta + alpha)) + 2.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (alpha + beta) + 2.0;
	tmp = 0.0;
	if (beta <= 9.2e+14)
		tmp = (((1.0 + beta) / (2.0 + beta)) / (3.0 + beta)) / t_0;
	else
		tmp = ((alpha - -1.0) / t_0) / ((1.0 + (beta + alpha)) + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 9.2e+14], N[(N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(3.0 + beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(1.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.2e14

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{1 + \beta}}{2 + \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. lower-+.f64 86.8

         \[\leadsto \frac{\frac{\frac{1 + \beta}{\color{blue}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Applied rewrites 86.8%

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{1 + \beta}{2 + \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1 + \beta}{2 + \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1 + \beta}{2 + \beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)}} \]

   7. Applied rewrites 86.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\beta - -1}{\beta + 2}}{3 + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2}} \]

   8. Taylor expanded in alpha around 0

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}}}{\left(\alpha + \beta\right) + 2} \]
   9. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1 + \beta}{2 + \beta}}{3 + \beta}}}{\left(\alpha + \beta\right) + 2} \]

      2. div-add-rev N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{2 + \beta} + \frac{\beta}{2 + \beta}}}{3 + \beta}}{\left(\alpha + \beta\right) + 2} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2 + \beta} + \frac{\beta}{2 + \beta}}{3 + \beta}}}{\left(\alpha + \beta\right) + 2} \]

      4. div-add-rev N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{3 + \beta}}{\left(\alpha + \beta\right) + 2} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{3 + \beta}}{\left(\alpha + \beta\right) + 2} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{1 + \beta}}{2 + \beta}}{3 + \beta}}{\left(\alpha + \beta\right) + 2} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{1 + \beta}{\color{blue}{2 + \beta}}}{3 + \beta}}{\left(\alpha + \beta\right) + 2} \]

      8. lower-+.f64 66.8

         \[\leadsto \frac{\frac{\frac{1 + \beta}{2 + \beta}}{\color{blue}{3 + \beta}}}{\left(\alpha + \beta\right) + 2} \]

   10. Applied rewrites 66.8%

       \[\leadsto \frac{\color{blue}{\frac{\frac{1 + \beta}{2 + \beta}}{3 + \beta}}}{\left(\alpha + \beta\right) + 2} \]

   if 9.2e14 < beta

   1. Initial program 79.4%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{1 + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1 + \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      6. lower-+.f64 79.4

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right)} + 2 \cdot 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\alpha + \beta\right)}\right) + 2 \cdot 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      9. lower-+.f64 79.4

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2 \cdot 1}} \]

      11. metadata-eval 79.4

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2}} \]

   4. Applied rewrites 79.4%

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\beta + \alpha\right)\right) + 2}} \]

   5. Taylor expanded in beta around -inf

      \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \left(-1 \cdot \alpha - 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{neg}\left(\left(-1 \cdot \alpha - 1\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{-\left(-1 \cdot \alpha - 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\frac{-\color{blue}{\left(-1 \cdot \alpha - 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\frac{-\left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} - 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      5. lower-neg.f64 83.7

         \[\leadsto \frac{\frac{-\left(\color{blue}{\left(-\alpha\right)} - 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   7. Applied rewrites 83.7%

      \[\leadsto \frac{\frac{\color{blue}{-\left(\left(-\alpha\right) - 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{\frac{1 + \beta}{2 + \beta}}{3 + \beta}}{\left(\alpha + \beta\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\alpha + \beta\right) + 2}}{\left(1 + \left(\beta + \alpha\right)\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 98.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 9.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{\frac{1 + \beta}{2 + \beta}}{3 + \beta}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{3 + \left(\alpha + \beta\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)))
   (if (<= beta 9.2e+14)
     (/ (/ (/ (+ 1.0 beta) (+ 2.0 beta)) (+ 3.0 beta)) t_0)
     (/ (/ (- alpha -1.0) (+ 3.0 (+ alpha beta))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 9.2e+14) {
		tmp = (((1.0 + beta) / (2.0 + beta)) / (3.0 + beta)) / t_0;
	} else {
		tmp = ((alpha - -1.0) / (3.0 + (alpha + beta))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (alpha + beta) + 2.0d0
    if (beta <= 9.2d+14) then
        tmp = (((1.0d0 + beta) / (2.0d0 + beta)) / (3.0d0 + beta)) / t_0
    else
        tmp = ((alpha - (-1.0d0)) / (3.0d0 + (alpha + beta))) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 9.2e+14) {
		tmp = (((1.0 + beta) / (2.0 + beta)) / (3.0 + beta)) / t_0;
	} else {
		tmp = ((alpha - -1.0) / (3.0 + (alpha + beta))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (alpha + beta) + 2.0
	tmp = 0
	if beta <= 9.2e+14:
		tmp = (((1.0 + beta) / (2.0 + beta)) / (3.0 + beta)) / t_0
	else:
		tmp = ((alpha - -1.0) / (3.0 + (alpha + beta))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (beta <= 9.2e+14)
		tmp = Float64(Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(3.0 + beta)) / t_0);
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / Float64(3.0 + Float64(alpha + beta))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (alpha + beta) + 2.0;
	tmp = 0.0;
	if (beta <= 9.2e+14)
		tmp = (((1.0 + beta) / (2.0 + beta)) / (3.0 + beta)) / t_0;
	else
		tmp = ((alpha - -1.0) / (3.0 + (alpha + beta))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 9.2e+14], N[(N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(3.0 + beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.2e14

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{1 + \beta}}{2 + \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. lower-+.f64 86.8

         \[\leadsto \frac{\frac{\frac{1 + \beta}{\color{blue}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Applied rewrites 86.8%

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{1 + \beta}{2 + \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1 + \beta}{2 + \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1 + \beta}{2 + \beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)}} \]

   7. Applied rewrites 86.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\beta - -1}{\beta + 2}}{3 + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2}} \]

   8. Taylor expanded in alpha around 0

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}}}{\left(\alpha + \beta\right) + 2} \]
   9. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1 + \beta}{2 + \beta}}{3 + \beta}}}{\left(\alpha + \beta\right) + 2} \]

      2. div-add-rev N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{2 + \beta} + \frac{\beta}{2 + \beta}}}{3 + \beta}}{\left(\alpha + \beta\right) + 2} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2 + \beta} + \frac{\beta}{2 + \beta}}{3 + \beta}}}{\left(\alpha + \beta\right) + 2} \]

      4. div-add-rev N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{3 + \beta}}{\left(\alpha + \beta\right) + 2} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{3 + \beta}}{\left(\alpha + \beta\right) + 2} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{1 + \beta}}{2 + \beta}}{3 + \beta}}{\left(\alpha + \beta\right) + 2} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{1 + \beta}{\color{blue}{2 + \beta}}}{3 + \beta}}{\left(\alpha + \beta\right) + 2} \]

      8. lower-+.f64 66.8

         \[\leadsto \frac{\frac{\frac{1 + \beta}{2 + \beta}}{\color{blue}{3 + \beta}}}{\left(\alpha + \beta\right) + 2} \]

   10. Applied rewrites 66.8%

       \[\leadsto \frac{\color{blue}{\frac{\frac{1 + \beta}{2 + \beta}}{3 + \beta}}}{\left(\alpha + \beta\right) + 2} \]

   if 9.2e14 < beta

   1. Initial program 79.4%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{1 + \beta}}{2 + \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. lower-+.f64 81.6

         \[\leadsto \frac{\frac{\frac{1 + \beta}{\color{blue}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Applied rewrites 81.6%

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{1 + \beta}{2 + \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1 + \beta}{2 + \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1 + \beta}{2 + \beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)}} \]

   7. Applied rewrites 81.6%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\beta - -1}{\beta + 2}}{3 + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2}} \]

   8. Taylor expanded in beta around -inf

      \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \left(-1 \cdot \alpha - 1\right)}}{3 + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{neg}\left(\left(-1 \cdot \alpha - 1\right)\right)}}{3 + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2} \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{-\left(-1 \cdot \alpha - 1\right)}}{3 + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\frac{-\color{blue}{\left(-1 \cdot \alpha - 1\right)}}{3 + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\frac{-\left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} - 1\right)}{3 + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2} \]

      5. lower-neg.f64 83.7

         \[\leadsto \frac{\frac{-\left(\color{blue}{\left(-\alpha\right)} - 1\right)}{3 + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2} \]

   10. Applied rewrites 83.7%

       \[\leadsto \frac{\frac{\color{blue}{-\left(\left(-\alpha\right) - 1\right)}}{3 + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{\frac{1 + \beta}{2 + \beta}}{3 + \beta}}{\left(\alpha + \beta\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{3 + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 98.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{\beta - -1}{\beta + 2}}{\mathsf{fma}\left(5 + \beta, \beta, 6\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{3 + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.5e+23)
   (/ (/ (- beta -1.0) (+ beta 2.0)) (fma (+ 5.0 beta) beta 6.0))
   (/ (/ (- alpha -1.0) (+ 3.0 (+ alpha beta))) (+ (+ alpha beta) 2.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.5e+23) {
		tmp = ((beta - -1.0) / (beta + 2.0)) / fma((5.0 + beta), beta, 6.0);
	} else {
		tmp = ((alpha - -1.0) / (3.0 + (alpha + beta))) / ((alpha + beta) + 2.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.5e+23)
		tmp = Float64(Float64(Float64(beta - -1.0) / Float64(beta + 2.0)) / fma(Float64(5.0 + beta), beta, 6.0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / Float64(3.0 + Float64(alpha + beta))) / Float64(Float64(alpha + beta) + 2.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 7.5e+23], N[(N[(N[(beta - -1.0), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(5.0 + beta), $MachinePrecision] * beta + 6.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.49999999999999987e23

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{1 + \beta}}{2 + \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. lower-+.f64 86.9

         \[\leadsto \frac{\frac{\frac{1 + \beta}{\color{blue}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Applied rewrites 86.9%

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   7. Applied rewrites 86.9%

      \[\leadsto \color{blue}{\frac{\frac{\beta - -1}{\beta + 2}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}} \]

   8. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{\beta - -1}{\beta + 2}}{\color{blue}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\beta - -1}{\beta + 2}}{\color{blue}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\beta - -1}{\beta + 2}}{\color{blue}{\left(2 + \beta\right)} \cdot \left(3 + \beta\right)} \]

      3. lower-+.f64 66.2

         \[\leadsto \frac{\frac{\beta - -1}{\beta + 2}}{\left(2 + \beta\right) \cdot \color{blue}{\left(3 + \beta\right)}} \]

   10. Applied rewrites 66.2%

       \[\leadsto \frac{\frac{\beta - -1}{\beta + 2}}{\color{blue}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]

   11. Taylor expanded in beta around 0

       \[\leadsto \frac{\frac{\beta - -1}{\beta + 2}}{6 + \color{blue}{\beta \cdot \left(5 + \beta\right)}} \]
   12. Step-by-step derivation

   13. Applied rewrites 66.2%

       \[\leadsto \frac{\frac{\beta - -1}{\beta + 2}}{\mathsf{fma}\left(5 + \beta, \color{blue}{\beta}, 6\right)} \]

   if 7.49999999999999987e23 < beta

   1. Initial program 78.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{1 + \beta}}{2 + \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. lower-+.f64 81.2

         \[\leadsto \frac{\frac{\frac{1 + \beta}{\color{blue}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Applied rewrites 81.2%

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{1 + \beta}{2 + \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1 + \beta}{2 + \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1 + \beta}{2 + \beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)}} \]

   7. Applied rewrites 81.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\beta - -1}{\beta + 2}}{3 + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2}} \]

   8. Taylor expanded in beta around -inf

      \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \left(-1 \cdot \alpha - 1\right)}}{3 + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{neg}\left(\left(-1 \cdot \alpha - 1\right)\right)}}{3 + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2} \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{-\left(-1 \cdot \alpha - 1\right)}}{3 + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\frac{-\color{blue}{\left(-1 \cdot \alpha - 1\right)}}{3 + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\frac{-\left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} - 1\right)}{3 + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2} \]

      5. lower-neg.f64 83.3

         \[\leadsto \frac{\frac{-\left(\color{blue}{\left(-\alpha\right)} - 1\right)}{3 + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2} \]

   10. Applied rewrites 83.3%

       \[\leadsto \frac{\frac{\color{blue}{-\left(\left(-\alpha\right) - 1\right)}}{3 + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{\beta - -1}{\beta + 2}}{\mathsf{fma}\left(5 + \beta, \beta, 6\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{3 + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 98.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+24}:\\ \;\;\;\;\frac{\frac{\beta - -1}{\beta + 2}}{\mathsf{fma}\left(5 + \beta, \beta, 6\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(\alpha + \beta\right) + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.2e+24)
   (/ (/ (- beta -1.0) (+ beta 2.0)) (fma (+ 5.0 beta) beta 6.0))
   (/ (/ (+ 1.0 alpha) beta) (+ (+ alpha beta) 2.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.2e+24) {
		tmp = ((beta - -1.0) / (beta + 2.0)) / fma((5.0 + beta), beta, 6.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / ((alpha + beta) + 2.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.2e+24)
		tmp = Float64(Float64(Float64(beta - -1.0) / Float64(beta + 2.0)) / fma(Float64(5.0 + beta), beta, 6.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(Float64(alpha + beta) + 2.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.2e+24], N[(N[(N[(beta - -1.0), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(5.0 + beta), $MachinePrecision] * beta + 6.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.1999999999999997e24

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{1 + \beta}}{2 + \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. lower-+.f64 86.9

         \[\leadsto \frac{\frac{\frac{1 + \beta}{\color{blue}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Applied rewrites 86.9%

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   7. Applied rewrites 86.9%

      \[\leadsto \color{blue}{\frac{\frac{\beta - -1}{\beta + 2}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}} \]

   8. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{\beta - -1}{\beta + 2}}{\color{blue}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\beta - -1}{\beta + 2}}{\color{blue}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\beta - -1}{\beta + 2}}{\color{blue}{\left(2 + \beta\right)} \cdot \left(3 + \beta\right)} \]

      3. lower-+.f64 66.2

         \[\leadsto \frac{\frac{\beta - -1}{\beta + 2}}{\left(2 + \beta\right) \cdot \color{blue}{\left(3 + \beta\right)}} \]

   10. Applied rewrites 66.2%

       \[\leadsto \frac{\frac{\beta - -1}{\beta + 2}}{\color{blue}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]

   11. Taylor expanded in beta around 0

       \[\leadsto \frac{\frac{\beta - -1}{\beta + 2}}{6 + \color{blue}{\beta \cdot \left(5 + \beta\right)}} \]
   12. Step-by-step derivation

   13. Applied rewrites 66.2%

       \[\leadsto \frac{\frac{\beta - -1}{\beta + 2}}{\mathsf{fma}\left(5 + \beta, \color{blue}{\beta}, 6\right)} \]

   if 3.1999999999999997e24 < beta

   1. Initial program 78.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{1 + \beta}}{2 + \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. lower-+.f64 81.2

         \[\leadsto \frac{\frac{\frac{1 + \beta}{\color{blue}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Applied rewrites 81.2%

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{1 + \beta}{2 + \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1 + \beta}{2 + \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1 + \beta}{2 + \beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)}} \]

   7. Applied rewrites 81.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\beta - -1}{\beta + 2}}{3 + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2}} \]

   8. Taylor expanded in beta around inf

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\alpha + \beta\right) + 2} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\alpha + \beta\right) + 2} \]

      2. lower-+.f64 82.8

         \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\beta}}{\left(\alpha + \beta\right) + 2} \]

   10. Applied rewrites 82.8%

       \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\alpha + \beta\right) + 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+24}:\\ \;\;\;\;\frac{\frac{\beta - -1}{\beta + 2}}{\mathsf{fma}\left(5 + \beta, \beta, 6\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(\alpha + \beta\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 97.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{\frac{\beta - -1}{\beta + 2}}{\mathsf{fma}\left(5, \beta, 6\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(1 + \left(\beta + \alpha\right)\right) + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.5)
   (/ (/ (- beta -1.0) (+ beta 2.0)) (fma 5.0 beta 6.0))
   (/ (/ (+ 1.0 alpha) beta) (+ (+ 1.0 (+ beta alpha)) 2.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5) {
		tmp = ((beta - -1.0) / (beta + 2.0)) / fma(5.0, beta, 6.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / ((1.0 + (beta + alpha)) + 2.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.5)
		tmp = Float64(Float64(Float64(beta - -1.0) / Float64(beta + 2.0)) / fma(5.0, beta, 6.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(Float64(1.0 + Float64(beta + alpha)) + 2.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.5], N[(N[(N[(beta - -1.0), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(5.0 * beta + 6.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(1.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{1 + \beta}}{2 + \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. lower-+.f64 87.0

         \[\leadsto \frac{\frac{\frac{1 + \beta}{\color{blue}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Applied rewrites 87.0%

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   7. Applied rewrites 87.0%

      \[\leadsto \color{blue}{\frac{\frac{\beta - -1}{\beta + 2}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}} \]

   8. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{\beta - -1}{\beta + 2}}{\color{blue}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\beta - -1}{\beta + 2}}{\color{blue}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\beta - -1}{\beta + 2}}{\color{blue}{\left(2 + \beta\right)} \cdot \left(3 + \beta\right)} \]

      3. lower-+.f64 65.8

         \[\leadsto \frac{\frac{\beta - -1}{\beta + 2}}{\left(2 + \beta\right) \cdot \color{blue}{\left(3 + \beta\right)}} \]

   10. Applied rewrites 65.8%

       \[\leadsto \frac{\frac{\beta - -1}{\beta + 2}}{\color{blue}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]

   11. Taylor expanded in beta around 0

       \[\leadsto \frac{\frac{\beta - -1}{\beta + 2}}{6 + \color{blue}{5 \cdot \beta}} \]
   12. Step-by-step derivation

   13. Applied rewrites 65.8%

       \[\leadsto \frac{\frac{\beta - -1}{\beta + 2}}{\mathsf{fma}\left(5, \color{blue}{\beta}, 6\right)} \]

   if 4.5 < beta

   1. Initial program 80.2%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{1 + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1 + \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      6. lower-+.f64 80.2

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right)} + 2 \cdot 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\alpha + \beta\right)}\right) + 2 \cdot 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      9. lower-+.f64 80.2

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2 \cdot 1}} \]

      11. metadata-eval 80.2

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2}} \]

   4. Applied rewrites 80.2%

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\beta + \alpha\right)\right) + 2}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      3. div-add N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} + \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      4. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\alpha + \beta\right) + 2 \cdot 1}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{1}{\left(\beta + \alpha\right) + \color{blue}{2}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      11. flip-+ N/A

          \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\frac{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2 \cdot 2}{\left(\beta + \alpha\right) - 2}}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      12. associate-/r/ N/A

          \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2 \cdot 2} \cdot \left(\left(\beta + \alpha\right) - 2\right)} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2 \cdot 2}, \left(\beta + \alpha\right) - 2, \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   6. Applied rewrites 80.2%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{{\left(\alpha + \beta\right)}^{2} - 4}, \left(\alpha + \beta\right) - 2, \frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right) + \frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{\alpha \cdot \beta + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      5. div-add N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\frac{\alpha \cdot \beta}{\left(\alpha + \beta\right) + 2} + \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2}\right)} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      6. associate-+l+ N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha \cdot \beta}{\left(\alpha + \beta\right) + 2} + \left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{\beta \cdot \alpha}}{\left(\alpha + \beta\right) + 2} + \left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      8. associate-/l* N/A

         \[\leadsto \frac{\frac{\color{blue}{\beta \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}} + \left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \color{blue}{\frac{\alpha}{\left(\alpha + \beta\right) + 2}}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \color{blue}{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   8. Applied rewrites 99.9%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + {\left(\left(\alpha + \beta\right) + 2\right)}^{-1}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   9. Taylor expanded in beta around inf

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

       2. lower-+.f64 81.4

          \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\beta}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   11. Applied rewrites 81.4%

       \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{\frac{\beta - -1}{\beta + 2}}{\mathsf{fma}\left(5, \beta, 6\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(1 + \left(\beta + \alpha\right)\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 97.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 82:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \beta, 0.5\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(1 + \left(\beta + \alpha\right)\right) + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 82.0)
   (/ (fma 0.25 beta 0.5) (* (+ (+ alpha beta) 2.0) (+ 3.0 (+ alpha beta))))
   (/ (/ (+ 1.0 alpha) beta) (+ (+ 1.0 (+ beta alpha)) 2.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 82.0) {
		tmp = fma(0.25, beta, 0.5) / (((alpha + beta) + 2.0) * (3.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / beta) / ((1.0 + (beta + alpha)) + 2.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 82.0)
		tmp = Float64(fma(0.25, beta, 0.5) / Float64(Float64(Float64(alpha + beta) + 2.0) * Float64(3.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(Float64(1.0 + Float64(beta + alpha)) + 2.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 82.0], N[(N[(0.25 * beta + 0.5), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision] * N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(1.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 82

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{1 + \beta}}{2 + \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. lower-+.f64 87.0

         \[\leadsto \frac{\frac{\frac{1 + \beta}{\color{blue}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Applied rewrites 87.0%

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   7. Applied rewrites 87.0%

      \[\leadsto \color{blue}{\frac{\frac{\beta - -1}{\beta + 2}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}} \]

   8. Taylor expanded in beta around 0

      \[\leadsto \frac{\frac{1}{2} + \color{blue}{\frac{1}{4} \cdot \beta}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 87.0%

       \[\leadsto \frac{\mathsf{fma}\left(0.25, \color{blue}{\beta}, 0.5\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)} \]

   if 82 < beta

   1. Initial program 80.2%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{1 + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1 + \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      6. lower-+.f64 80.2

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right)} + 2 \cdot 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\alpha + \beta\right)}\right) + 2 \cdot 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      9. lower-+.f64 80.2

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2 \cdot 1}} \]

      11. metadata-eval 80.2

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2}} \]

   4. Applied rewrites 80.2%

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\beta + \alpha\right)\right) + 2}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      3. div-add N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} + \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      4. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\alpha + \beta\right) + 2 \cdot 1}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{1}{\left(\beta + \alpha\right) + \color{blue}{2}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      11. flip-+ N/A

          \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\frac{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2 \cdot 2}{\left(\beta + \alpha\right) - 2}}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      12. associate-/r/ N/A

          \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2 \cdot 2} \cdot \left(\left(\beta + \alpha\right) - 2\right)} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2 \cdot 2}, \left(\beta + \alpha\right) - 2, \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   6. Applied rewrites 80.2%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{{\left(\alpha + \beta\right)}^{2} - 4}, \left(\alpha + \beta\right) - 2, \frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right) + \frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{\alpha \cdot \beta + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      5. div-add N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\frac{\alpha \cdot \beta}{\left(\alpha + \beta\right) + 2} + \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2}\right)} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      6. associate-+l+ N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha \cdot \beta}{\left(\alpha + \beta\right) + 2} + \left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{\beta \cdot \alpha}}{\left(\alpha + \beta\right) + 2} + \left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      8. associate-/l* N/A

         \[\leadsto \frac{\frac{\color{blue}{\beta \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}} + \left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \color{blue}{\frac{\alpha}{\left(\alpha + \beta\right) + 2}}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \color{blue}{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   8. Applied rewrites 99.9%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + {\left(\left(\alpha + \beta\right) + 2\right)}^{-1}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   9. Taylor expanded in beta around inf

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

       2. lower-+.f64 81.4

          \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\beta}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   11. Applied rewrites 81.4%

       \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 96.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 82:\\ \;\;\;\;\frac{0.5}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(1 + \left(\beta + \alpha\right)\right) + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 82.0)
   (/ 0.5 (* (+ 3.0 alpha) (+ 2.0 alpha)))
   (/ (/ (+ 1.0 alpha) beta) (+ (+ 1.0 (+ beta alpha)) 2.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 82.0) {
		tmp = 0.5 / ((3.0 + alpha) * (2.0 + alpha));
	} else {
		tmp = ((1.0 + alpha) / beta) / ((1.0 + (beta + alpha)) + 2.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 82.0d0) then
        tmp = 0.5d0 / ((3.0d0 + alpha) * (2.0d0 + alpha))
    else
        tmp = ((1.0d0 + alpha) / beta) / ((1.0d0 + (beta + alpha)) + 2.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 82.0) {
		tmp = 0.5 / ((3.0 + alpha) * (2.0 + alpha));
	} else {
		tmp = ((1.0 + alpha) / beta) / ((1.0 + (beta + alpha)) + 2.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 82.0:
		tmp = 0.5 / ((3.0 + alpha) * (2.0 + alpha))
	else:
		tmp = ((1.0 + alpha) / beta) / ((1.0 + (beta + alpha)) + 2.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 82.0)
		tmp = Float64(0.5 / Float64(Float64(3.0 + alpha) * Float64(2.0 + alpha)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(Float64(1.0 + Float64(beta + alpha)) + 2.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 82.0)
		tmp = 0.5 / ((3.0 + alpha) * (2.0 + alpha));
	else
		tmp = ((1.0 + alpha) / beta) / ((1.0 + (beta + alpha)) + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 82.0], N[(0.5 / N[(N[(3.0 + alpha), $MachinePrecision] * N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(1.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 82

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{1 + \beta}}{2 + \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. lower-+.f64 87.0

         \[\leadsto \frac{\frac{\frac{1 + \beta}{\color{blue}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Applied rewrites 87.0%

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   7. Applied rewrites 87.0%

      \[\leadsto \color{blue}{\frac{\frac{\beta - -1}{\beta + 2}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}} \]

   8. Taylor expanded in beta around 0

      \[\leadsto \frac{\frac{1}{2}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 86.3%

       \[\leadsto \frac{0.5}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)} \]

   11. Taylor expanded in beta around 0

       \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\left(3 + \alpha\right)} \cdot \left(2 + \alpha\right)} \]

       4. lower-+.f64 86.4

          \[\leadsto \frac{0.5}{\left(3 + \alpha\right) \cdot \color{blue}{\left(2 + \alpha\right)}} \]

   13. Applied rewrites 86.4%

       \[\leadsto \frac{0.5}{\color{blue}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}} \]

   if 82 < beta

   1. Initial program 80.2%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{1 + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1 + \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      6. lower-+.f64 80.2

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right)} + 2 \cdot 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\alpha + \beta\right)}\right) + 2 \cdot 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      9. lower-+.f64 80.2

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2 \cdot 1}} \]

      11. metadata-eval 80.2

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2}} \]

   4. Applied rewrites 80.2%

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\beta + \alpha\right)\right) + 2}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      3. div-add N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} + \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      4. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\alpha + \beta\right) + 2 \cdot 1}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot 1} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{1}{\left(\beta + \alpha\right) + \color{blue}{2}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      11. flip-+ N/A

          \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\frac{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2 \cdot 2}{\left(\beta + \alpha\right) - 2}}} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      12. associate-/r/ N/A

          \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2 \cdot 2} \cdot \left(\left(\beta + \alpha\right) - 2\right)} + \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) - 2 \cdot 2}, \left(\beta + \alpha\right) - 2, \frac{\left(\alpha + \beta\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   6. Applied rewrites 80.2%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{{\left(\alpha + \beta\right)}^{2} - 4}, \left(\alpha + \beta\right) - 2, \frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right) + \frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{\alpha \cdot \beta + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      5. div-add N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\frac{\alpha \cdot \beta}{\left(\alpha + \beta\right) + 2} + \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2}\right)} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      6. associate-+l+ N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha \cdot \beta}{\left(\alpha + \beta\right) + 2} + \left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{\beta \cdot \alpha}}{\left(\alpha + \beta\right) + 2} + \left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      8. associate-/l* N/A

         \[\leadsto \frac{\frac{\color{blue}{\beta \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}} + \left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \color{blue}{\frac{\alpha}{\left(\alpha + \beta\right) + 2}}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \color{blue}{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + \frac{1}{{\left(\alpha + \beta\right)}^{2} - 4} \cdot \left(\left(\alpha + \beta\right) - 2\right)}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   8. Applied rewrites 99.9%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\beta, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2} + {\left(\left(\alpha + \beta\right) + 2\right)}^{-1}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   9. Taylor expanded in beta around inf

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

       2. lower-+.f64 81.4

          \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\beta}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   11. Applied rewrites 81.4%

       \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 82:\\ \;\;\;\;\frac{0.5}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(1 + \left(\beta + \alpha\right)\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 96.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 82:\\ \;\;\;\;\frac{0.5}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(\alpha + \beta\right) + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 82.0)
   (/ 0.5 (* (+ 3.0 alpha) (+ 2.0 alpha)))
   (/ (/ (+ 1.0 alpha) beta) (+ (+ alpha beta) 2.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 82.0) {
		tmp = 0.5 / ((3.0 + alpha) * (2.0 + alpha));
	} else {
		tmp = ((1.0 + alpha) / beta) / ((alpha + beta) + 2.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 82.0d0) then
        tmp = 0.5d0 / ((3.0d0 + alpha) * (2.0d0 + alpha))
    else
        tmp = ((1.0d0 + alpha) / beta) / ((alpha + beta) + 2.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 82.0) {
		tmp = 0.5 / ((3.0 + alpha) * (2.0 + alpha));
	} else {
		tmp = ((1.0 + alpha) / beta) / ((alpha + beta) + 2.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 82.0:
		tmp = 0.5 / ((3.0 + alpha) * (2.0 + alpha))
	else:
		tmp = ((1.0 + alpha) / beta) / ((alpha + beta) + 2.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 82.0)
		tmp = Float64(0.5 / Float64(Float64(3.0 + alpha) * Float64(2.0 + alpha)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(Float64(alpha + beta) + 2.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 82.0)
		tmp = 0.5 / ((3.0 + alpha) * (2.0 + alpha));
	else
		tmp = ((1.0 + alpha) / beta) / ((alpha + beta) + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 82.0], N[(0.5 / N[(N[(3.0 + alpha), $MachinePrecision] * N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 82

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{1 + \beta}}{2 + \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. lower-+.f64 87.0

         \[\leadsto \frac{\frac{\frac{1 + \beta}{\color{blue}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Applied rewrites 87.0%

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   7. Applied rewrites 87.0%

      \[\leadsto \color{blue}{\frac{\frac{\beta - -1}{\beta + 2}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}} \]

   8. Taylor expanded in beta around 0

      \[\leadsto \frac{\frac{1}{2}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 86.3%

       \[\leadsto \frac{0.5}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)} \]

   11. Taylor expanded in beta around 0

       \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\left(3 + \alpha\right)} \cdot \left(2 + \alpha\right)} \]

       4. lower-+.f64 86.4

          \[\leadsto \frac{0.5}{\left(3 + \alpha\right) \cdot \color{blue}{\left(2 + \alpha\right)}} \]

   13. Applied rewrites 86.4%

       \[\leadsto \frac{0.5}{\color{blue}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}} \]

   if 82 < beta

   1. Initial program 80.2%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{1 + \beta}}{2 + \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. lower-+.f64 81.2

         \[\leadsto \frac{\frac{\frac{1 + \beta}{\color{blue}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Applied rewrites 81.2%

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{1 + \beta}{2 + \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1 + \beta}{2 + \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1 + \beta}{2 + \beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)}} \]

   7. Applied rewrites 81.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\beta - -1}{\beta + 2}}{3 + \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2}} \]

   8. Taylor expanded in beta around inf

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\alpha + \beta\right) + 2} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\alpha + \beta\right) + 2} \]

      2. lower-+.f64 81.4

         \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\beta}}{\left(\alpha + \beta\right) + 2} \]

   10. Applied rewrites 81.4%

       \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\alpha + \beta\right) + 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 82:\\ \;\;\;\;\frac{0.5}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(\alpha + \beta\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 96.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 82:\\ \;\;\;\;\frac{0.5}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}\\ \mathbf{elif}\;\beta \leq 9 \cdot 10^{+158}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 82.0)
   (/ 0.5 (* (+ 3.0 alpha) (+ 2.0 alpha)))
   (if (<= beta 9e+158)
     (/ (+ 1.0 alpha) (* beta beta))
     (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 82.0) {
		tmp = 0.5 / ((3.0 + alpha) * (2.0 + alpha));
	} else if (beta <= 9e+158) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 82.0d0) then
        tmp = 0.5d0 / ((3.0d0 + alpha) * (2.0d0 + alpha))
    else if (beta <= 9d+158) then
        tmp = (1.0d0 + alpha) / (beta * beta)
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 82.0) {
		tmp = 0.5 / ((3.0 + alpha) * (2.0 + alpha));
	} else if (beta <= 9e+158) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 82.0:
		tmp = 0.5 / ((3.0 + alpha) * (2.0 + alpha))
	elif beta <= 9e+158:
		tmp = (1.0 + alpha) / (beta * beta)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 82.0)
		tmp = Float64(0.5 / Float64(Float64(3.0 + alpha) * Float64(2.0 + alpha)));
	elseif (beta <= 9e+158)
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 82.0)
		tmp = 0.5 / ((3.0 + alpha) * (2.0 + alpha));
	elseif (beta <= 9e+158)
		tmp = (1.0 + alpha) / (beta * beta);
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 82.0], N[(0.5 / N[(N[(3.0 + alpha), $MachinePrecision] * N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 9e+158], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 82

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{1 + \beta}}{2 + \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. lower-+.f64 87.0

         \[\leadsto \frac{\frac{\frac{1 + \beta}{\color{blue}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Applied rewrites 87.0%

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   7. Applied rewrites 87.0%

      \[\leadsto \color{blue}{\frac{\frac{\beta - -1}{\beta + 2}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}} \]

   8. Taylor expanded in beta around 0

      \[\leadsto \frac{\frac{1}{2}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 86.3%

       \[\leadsto \frac{0.5}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)} \]

   11. Taylor expanded in beta around 0

       \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\left(3 + \alpha\right)} \cdot \left(2 + \alpha\right)} \]

       4. lower-+.f64 86.4

          \[\leadsto \frac{0.5}{\left(3 + \alpha\right) \cdot \color{blue}{\left(2 + \alpha\right)}} \]

   13. Applied rewrites 86.4%

       \[\leadsto \frac{0.5}{\color{blue}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}} \]

   if 82 < beta < 9.00000000000000092e158

   1. Initial program 83.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

      4. lower-*.f64 66.2

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   if 9.00000000000000092e158 < beta

   1. Initial program 77.6%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

      4. lower-*.f64 86.1

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 86.1%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   6. Taylor expanded in alpha around inf

      \[\leadsto \frac{\alpha}{\color{blue}{{\beta}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 86.1%

      \[\leadsto \frac{\alpha}{\color{blue}{\beta \cdot \beta}} \]
   9. Step-by-step derivation

   10. Applied rewrites 91.0%

       \[\leadsto \frac{\frac{\alpha}{\beta}}{\beta} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 82:\\ \;\;\;\;\frac{0.5}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}\\ \mathbf{elif}\;\beta \leq 9 \cdot 10^{+158}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 96.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 82:\\ \;\;\;\;\frac{0.5}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 82.0)
   (/ 0.5 (* (+ 3.0 alpha) (+ 2.0 alpha)))
   (/ (/ (- alpha -1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 82.0) {
		tmp = 0.5 / ((3.0 + alpha) * (2.0 + alpha));
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 82.0d0) then
        tmp = 0.5d0 / ((3.0d0 + alpha) * (2.0d0 + alpha))
    else
        tmp = ((alpha - (-1.0d0)) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 82.0) {
		tmp = 0.5 / ((3.0 + alpha) * (2.0 + alpha));
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 82.0:
		tmp = 0.5 / ((3.0 + alpha) * (2.0 + alpha))
	else:
		tmp = ((alpha - -1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 82.0)
		tmp = Float64(0.5 / Float64(Float64(3.0 + alpha) * Float64(2.0 + alpha)));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 82.0)
		tmp = 0.5 / ((3.0 + alpha) * (2.0 + alpha));
	else
		tmp = ((alpha - -1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 82.0], N[(0.5 / N[(N[(3.0 + alpha), $MachinePrecision] * N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 82

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{1 + \beta}}{2 + \beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. lower-+.f64 87.0

         \[\leadsto \frac{\frac{\frac{1 + \beta}{\color{blue}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Applied rewrites 87.0%

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   7. Applied rewrites 87.0%

      \[\leadsto \color{blue}{\frac{\frac{\beta - -1}{\beta + 2}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}} \]

   8. Taylor expanded in beta around 0

      \[\leadsto \frac{\frac{1}{2}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 86.3%

       \[\leadsto \frac{0.5}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)} \]

   11. Taylor expanded in beta around 0

       \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\left(3 + \alpha\right)} \cdot \left(2 + \alpha\right)} \]

       4. lower-+.f64 86.4

          \[\leadsto \frac{0.5}{\left(3 + \alpha\right) \cdot \color{blue}{\left(2 + \alpha\right)}} \]

   13. Applied rewrites 86.4%

       \[\leadsto \frac{0.5}{\color{blue}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}} \]

   if 82 < beta

   1. Initial program 80.2%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

      4. lower-*.f64 77.9

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 77.9%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
   6. Step-by-step derivation

   7. Applied rewrites 81.2%

      \[\leadsto \color{blue}{\frac{\frac{\alpha - -1}{\beta}}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 82:\\ \;\;\;\;\frac{0.5}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 55.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 56000000:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 56000000.0)
   (/ (+ 1.0 alpha) (* beta beta))
   (/ (/ alpha beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 56000000.0) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 56000000.0d0) then
        tmp = (1.0d0 + alpha) / (beta * beta)
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 56000000.0) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if alpha <= 56000000.0:
		tmp = (1.0 + alpha) / (beta * beta)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 56000000.0)
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 56000000.0)
		tmp = (1.0 + alpha) / (beta * beta);
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[alpha, 56000000.0], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 5.6e7

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

      4. lower-*.f64 31.5

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 31.5%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   if 5.6e7 < alpha

   1. Initial program 83.3%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

      4. lower-*.f64 13.5

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 13.5%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   6. Taylor expanded in alpha around inf

      \[\leadsto \frac{\alpha}{\color{blue}{{\beta}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 13.5%

      \[\leadsto \frac{\alpha}{\color{blue}{\beta \cdot \beta}} \]
   9. Step-by-step derivation

   10. Applied rewrites 15.9%

       \[\leadsto \frac{\frac{\alpha}{\beta}}{\beta} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 52.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.0) (/ 1.0 (* beta beta)) (/ alpha (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.0) {
		tmp = 1.0 / (beta * beta);
	} else {
		tmp = alpha / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1.0d0) then
        tmp = 1.0d0 / (beta * beta)
    else
        tmp = alpha / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.0) {
		tmp = 1.0 / (beta * beta);
	} else {
		tmp = alpha / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if alpha <= 1.0:
		tmp = 1.0 / (beta * beta)
	else:
		tmp = alpha / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.0)
		tmp = Float64(1.0 / Float64(beta * beta));
	else
		tmp = Float64(alpha / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1.0)
		tmp = 1.0 / (beta * beta);
	else
		tmp = alpha / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[alpha, 1.0], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(alpha / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

      4. lower-*.f64 31.1

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 31.1%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{1}{\color{blue}{\beta} \cdot \beta} \]
   7. Step-by-step derivation

   8. Applied rewrites 30.8%

      \[\leadsto \frac{1}{\color{blue}{\beta} \cdot \beta} \]

   if 1 < alpha

   1. Initial program 83.5%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

      4. lower-*.f64 14.5

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 14.5%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   6. Taylor expanded in alpha around inf

      \[\leadsto \frac{\alpha}{\color{blue}{{\beta}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 14.5%

      \[\leadsto \frac{\alpha}{\color{blue}{\beta \cdot \beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 53.2% accurate, 4.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{1 + \alpha}{\beta \cdot \beta} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ (+ 1.0 alpha) (* beta beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return (1.0 + alpha) / (beta * beta);
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = (1.0d0 + alpha) / (beta * beta)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return (1.0 + alpha) / (beta * beta);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return (1.0 + alpha) / (beta * beta)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(Float64(1.0 + alpha) / Float64(beta * beta))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = (1.0 + alpha) / (beta * beta);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.1%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Add Preprocessing

3. Taylor expanded in beta around inf

   \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

   3. unpow2 N/A

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   4. lower-*.f64 25.2

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

5. Applied rewrites 25.2%

   \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
6. Add Preprocessing

Alternative 25: 32.1% accurate, 4.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{\alpha}{\beta \cdot \beta} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ alpha (* beta beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return alpha / (beta * beta);
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = alpha / (beta * beta)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return alpha / (beta * beta);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return alpha / (beta * beta)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(alpha / Float64(beta * beta))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = alpha / (beta * beta);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(alpha / N[(beta * beta), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.1%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Add Preprocessing

3. Taylor expanded in beta around inf

   \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

   3. unpow2 N/A

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   4. lower-*.f64 25.2

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

5. Applied rewrites 25.2%

   \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

6. Taylor expanded in alpha around inf

   \[\leadsto \frac{\alpha}{\color{blue}{{\beta}^{2}}} \]
7. Step-by-step derivation

8. Applied rewrites 18.6%

   \[\leadsto \frac{\alpha}{\color{blue}{\beta \cdot \beta}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025010 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))